A Stationarity-and-Coupling Criterion for Training-Free Time-Lagged Spectral Embeddings of Multivariate Time Series

# A Stationarity-and-Coupling Criterion for Training-Free Time-Lagged Spectral Embeddings of Multivariate Time Series
Source: [https://arxiv.org/html/2606.13823](https://arxiv.org/html/2606.13823)
###### Abstract

We study training\-free fixed\-length descriptors for multivariate time series and ask a question the representation\-learning literature usually leaves implicit: not merely whether such a descriptor performs well on a benchmark, but*when*it can be expected to work at all\. Our object of study isD​\(τ\)D\(\\tau\), a descriptor built from the time\-lagged correlation matrix that one of us developed for random\-matrix monitoring of network traffic\[[1](https://arxiv.org/html/2606.13823#bib.bib1),[2](https://arxiv.org/html/2606.13823#bib.bib2),[3](https://arxiv.org/html/2606.13823#bib.bib3)\], truncated at the Marchenko–Pastur \(MP\) edge so that only signal\-bearing eigenvalues survive, and classified by cosine similarity to class centroids with zero learned parameters\. The central contribution is not the descriptor but a*falsifiable applicability criterion*for it\. Working from a stationary Gaussian VAR\(1\) generative model, we argue thatD​\(τ\)D\(\\tau\)separates two classes when the signals are approximately stationary and the class information resides in their cross\-channel*temporal coupling*rather than in marginal per\-channel power\. We derive, semi\-formally, three consequences of this model: a condition under which two classes are distinguishable in the embedded space, why the static \(τ=0\\tau\{=\}0\) covariance alone collapses to chance, and why a stationary but power\-discriminated paradigm defeats the descriptor even though every sample is well\-conditioned\. The criterion is operational: a two\-part pre\-flight test, an augmented Dickey–Fuller stationarity check together with a power\-baseline saturation check, predicts applicability before any training, and we summarize it as a decision rule\. We validate both halves of the boundary on a deliberately mixed assortment\. On four paradigms that satisfy the criterion — Sleep\-EDF sleep staging, BCI\-IV\-2a motor imagery, MIT\-BIH arrhythmia, and ESC\-50 environmental sound — the descriptor is competitive with strong baselines at a fraction of their cost, reaching88\.5±4\.5%88\.5\\pm 4\.5\\%under 20\-subject leave\-one\-subject\-out on Sleep\-EDF in about thirteen minutes on a single CPU thread with no GPU\. On three paradigms that violate it — transient\-response ERPs, which are non\-stationary, and financial\-volatility regimes together with wearable stress detection, both power\-discriminated — the descriptor fails the criterion, collapsing to chance on the non\-stationary case and being outperformed by a simple power baseline on the power\-discriminated ones, exactly as the pre\-flight test anticipates; these negative results are the more informative half of the story\. We are explicit throughout thatD​\(τ\)D\(\\tau\)is not the most accurate representation available; its value is a compact, training\-free embedding whose domain of validity is known in advance\.

## 1Introduction

A multivariate time series, no matter what sensor produced it, carries its dynamics in the way its channels co\-vary across time\. A camera, a scalp electrode array, a microphone, and a chest lead all return the same mathematical object, a real matrix whose rows index time and whose columns index channels, and the discriminative content that distinguishes one regime from another is written into the second\-order temporal\-coupling structure of that matrix\. Most of the machinery the field brings to bear on such data, however, is modality\-specific by construction: a video model assumes a spatial grid, a sleep stager assumes electroencephalography, an acoustic model assumes a spectrogram front\-end\. Each is excellent in its lane and silent outside it\. We are interested in the opposite design point, a single descriptor that is computed identically for every modality, carries no learned parameters, and reduces any windowed multivariate signal to a fixed\-length vector that one may compare by cosine similarity\.

The appeal of a training\-free descriptor is sharpest precisely where the deep\-learning recipe is hardest to apply\. Many deployments require comparing or classifying multivariate signals with little or no labeled training data and without a GPU: on\-device triage, sleep\-stage or drowsiness monitoring, brain\-computer\-interface calibration for a new subject, or machine\-health monitoring under shifting operating conditions\. Deep models dominate the leaderboards on each of these in isolation, but they assume labeled in\-distribution corpora, accelerator hardware, and a training loop, and those assumptions fail jointly in the regimes above\. A descriptor that requires none of them is therefore useful both as a strong zero\-training baseline and as a feature extractor that a small calibration head can sit on top of\.

We can categorise the existing training\-free descriptors for sequential data by the spectrum they expose and the way they choose a rank cutoff\. A first family fits an explicit generative model and reads off its parameters: the dynamic\-texture linear dynamical systems of Doretto et al\.\[[11](https://arxiv.org/html/2606.13823#bib.bib11)\]regress a state transition matrix from a frame\-stack singular value decomposition and use its eigenvalues as the descriptor\. A second family applies a fixed analytic operator: the HiPPO polynomial\-projection operators of Gu et al\.\[[12](https://arxiv.org/html/2606.13823#bib.bib12)\]compress recent history into a coefficient vector that is optimal for a prescribed projection basis, the Fourier variant amounting to a sliding discrete Fourier transform\. A third family is spectral in the random\-matrix sense, ranking eigen\-components of a sample covariance and using the Marchenko–Pastur law to decide which of them are signal: the patch\-Casorati denoiser MPPCA of Veraart et al\.\[[9](https://arxiv.org/html/2606.13823#bib.bib9)\]and the time\-lagged correlation matrices that one of us developed for monitoring inter\-domain network traffic\[[1](https://arxiv.org/html/2606.13823#bib.bib1),[2](https://arxiv.org/html/2606.13823#bib.bib2),[3](https://arxiv.org/html/2606.13823#bib.bib3)\]both belong here\. The third family is attractive because, unlike variance\-ranked methods such as PCA or ICA that simply assume the high\-variance directions carry the dynamics, it uses the rigorous instruments of random\-matrix theory to identify the noise and to establish its spectral boundary, and then retains only the eigenvalues that provably exceed it\.

We take the time\-lagged correlation matrixD​\(τ\)D\(\\tau\)from that earlier network\-monitoring line of work and turn it into a general descriptor by pairing it with the Marchenko–Pastur edge\[[4](https://arxiv.org/html/2606.13823#bib.bib4),[5](https://arxiv.org/html/2606.13823#bib.bib5)\]as the noise cutoff\. Our central claim is not that this descriptor is the most accurate representation of multivariate time series — it is not — but that, unlike most representations, it comes with a precise and falsifiable account of*when*it is the right tool\. We make four claims\. First and foremost, we give an*applicability criterion*: working from a stationary Gaussian VAR\(1\) generative model, we show thatD​\(τ\)D\(\\tau\)separates two classes exactly when the signal is approximately stationary and the class information lives in cross\-channel temporal coupling rather than in marginal per\-channel power, and we render the criterion operational as a two\-part pre\-flight test that is checkable before any training\. Second, we confirm the positive half of the criterion on four paradigms that satisfy it — sleep staging, motor imagery, arrhythmia, and environmental sound — where the descriptor matches or beats channel\-power baselines, beating a trivial one outright and reaching parity with a strong multi\-band one, and is competitive with strong learned methods at a fraction of their compute\. Third, and most informatively, we confirm the negative half on three paradigms that violate it — transient\-response event\-related potentials, which are non\-stationary, and two stationary but power\-discriminated tasks, financial\-volatility regimes and wearable stress detection — on each of which the descriptor fails as the pre\-flight test predicts, and notably the criterion separates a physiological failure \(stress\) from the physiological successes\. Fourth, the descriptor is compact, training\-free, and noise\-invariant by construction, running in milliseconds per window on a single CPU core with no model file, GPU, or training step, and we release it as a single Python package\.

The remainder of the paper is organized as follows\. Section[2](https://arxiv.org/html/2606.13823#S2)situates the descriptor among training\-free spectral methods and the random\-matrix literature\. Section[3](https://arxiv.org/html/2606.13823#S3)definesD​\(τ\)D\(\\tau\)and the embedding pipeline\. Section[4](https://arxiv.org/html/2606.13823#S4), the heart of the paper, develops the applicability criterion: the VAR\(1\) identifiability argument, the two preconditions, the resulting decision rule, and the computational complexity\. Section[5](https://arxiv.org/html/2606.13823#S5)describes the benchmarks and baselines, and Section[6](https://arxiv.org/html/2606.13823#S6)reports point estimates with bootstrap confidence intervals on the four positive paradigms\. Section[7](https://arxiv.org/html/2606.13823#S7)confirms the two negative results that mark the reach of the descriptor, and Section[8](https://arxiv.org/html/2606.13823#S8)positions the method against learned heads and class\-aware specialists\. We conclude in Section[9](https://arxiv.org/html/2606.13823#S9)\.

## 2Related Work

The construction we extend is our own\. One of us introduced the symmetrised time\-lagged correlation matrix in a random\-matrix study of inter\-domain network traffic\[[1](https://arxiv.org/html/2606.13823#bib.bib1),[2](https://arxiv.org/html/2606.13823#bib.bib2),[3](https://arxiv.org/html/2606.13823#bib.bib3)\], where the point was that the spectrum of the lagged matrix separates a bulk of Marchenko–Pastur noise from a small number of outlier eigenvalues that encode the characteristic rhythms of the monitored system\. That construction is itself the natural lagged generalization of the “noise dressing” of equal\-time financial correlation matrices by Laloux et al\.\[[6](https://arxiv.org/html/2606.13823#bib.bib6)\]and Plerou et al\.\[[7](https://arxiv.org/html/2606.13823#bib.bib7)\], in which the Marchenko–Pastur edge marks the boundary above which an eigenvalue is inconsistent with pure noise\. The present paper carries that idea out of network monitoring and asks whether the same matrix, computed without modification, is a useful descriptor for video, audio, and physiological sensors\. To our knowledge the only prior use of a lagged random\-matrix construction on image data is the visual\-lifelog study of Li et al\.\[[15](https://arxiv.org/html/2606.13823#bib.bib15)\], which operated on a single grayscale stream and did not benchmark against learned baselines\.

The same Marchenko–Pastur cutoff underlies a second strand of random\-matrix work that we draw on for the noise threshold but not for the descriptor itself\. Veraart et al\.\[[9](https://arxiv.org/html/2606.13823#bib.bib9)\]popularised MPPCA as a patch\-Casorati denoiser for diffusion MRI, applying the MP edge to local patch covariance matrices and recovering the noise levelσ\\sigmafrom the bulk position as a free byproduct, and Cordero\-Grande et al\.\[[10](https://arxiv.org/html/2606.13823#bib.bib10)\]extended it to complex\-valued signals\. We use MPPCA only to confirm that the MP machinery transfers to image data; the descriptor we propose is the eigenstructure ofD​\(τ\)D\(\\tau\), not a denoised reconstruction\.

Two further families furnish the training\-free baselines against which we measure\. Doretto et al\.\[[11](https://arxiv.org/html/2606.13823#bib.bib11)\]fit a linear dynamical systemxt\+1=A​xt\+νt,yt=C​xt\+wtx\_\{t\+1\}\{=\}Ax\_\{t\}\+\\nu\_\{t\},\\;y\_\{t\}\{=\}Cx\_\{t\}\+w\_\{t\}to each video, takingCCfrom a frame\-stack singular value decomposition andAAfrom a least\-squares regression of the next state on the current one, and use the eigenvalues ofAAas the descriptor\. The construction is powerful on video but is intrinsically tied to spatial\-frame structure through the SVD step, and electrode channels possess no analogous spatial dimension, so the method is video\-specific\. The HiPPO operators of Gu et al\.\[[12](https://arxiv.org/html/2606.13823#bib.bib12)\]take the complementary route of a fixed analytic operator, deriving recurrent state\-space matrices\(A,B\)\(A,B\)that compress input history into a coefficient vector under a prescribed polynomial\-projection optimality; the Fourier\-boxcar variant is a true sliding\-window discrete Fourier transform realised as a continuous\-time linear ordinary differential equation withAApure\-imaginary and\|Ad\|=1\|A\_\{d\}\|\{=\}1\. A sibling research thread reports that this operator, followed by a single learned linear regression head, reaches∼\\sim93% on KTH 6\-class action recognition\. We evaluate the same operator under our pure cosine\-similarity protocol, with no learned head, so that the comparison is genuinely training\-free on both sides\.

For the supervised reference points we adopt the established specialists of each benchmark\. Random forests and support vector machines on hand\-crafted features reach roughly 70–75% on Sleep\-EDF 5\-class staging, while the deep models DeepSleepNet\[[18](https://arxiv.org/html/2606.13823#bib.bib18)\]and AttnSleep\[[23](https://arxiv.org/html/2606.13823#bib.bib23)\]reach 82–84%\. Common Spatial Patterns followed by linear discriminant analysis \(CSP\+LDA\) reach about 70% per subject on BCI\-IV\-2a 4\-class motor imagery, and deep methods exceed 80%\. We compare against CSP\+LDA and the channel power spectral density as the appropriate class\-aware and training\-free baselines, respectively\.

## 3Method

#### Setup\.

Letg∈ℝT×Ng\\in\\mathbb\{R\}^\{T\\times N\}be a multivariate time\-series matrix: rows index time, columns index channels \(e\.g\. EEG electrodes, ECG leads, or audio bands\)\. The construction treats any such matrix identically; sensors with a native channel count are used directly, and low\-channel sensors are augmented as described below\.

#### Preprocessing\.

Two operations remove a shared mode and drift\. The first is a*common\-average reference*\(CAR\): at each timestep we subtract the across\-channel sample mean, projecting out the dominant mode common to all channels\.111For image\-structured inputs this generalizes to a per\-frame rank\-1 SVD residual, the across\-channel mean being the rank\-1 mode of a one\-dimensional channel vector\.The second is a*per\-nodezz\-score*,gt​i←\(gt​i−g¯i\)/σig\_\{ti\}\\leftarrow\(g\_\{ti\}\{\-\}\\bar\{g\}\_\{i\}\)/\\sigma\_\{i\}, whereg¯i,σi\\bar\{g\}\_\{i\},\\sigma\_\{i\}are the per\-node temporal mean and standard deviation\. After preprocessing, each column ofgghas zero temporal mean and unit variance, while cross\-column means and variances are left unconstrained\.

#### Time\-lagged correlation matrix\.

Following Rojkova & Kantardzic\[[2](https://arxiv.org/html/2606.13823#bib.bib2)\], define the symmetrised lagged correlation

Di​j​\(τ\)=12​\(T−τ\)​∑t=1T−τ\[gi​\(t\)​gj​\(t\+τ\)\+gj​\(t\)​gi​\(t\+τ\)\]\.D\_\{ij\}\(\\tau\)\\;=\\;\\frac\{1\}\{2\(T\{\-\}\\tau\)\}\\sum\_\{t=1\}^\{T\-\\tau\}\\big\[g\_\{i\}\(t\)g\_\{j\}\(t\{\+\}\\tau\)\+g\_\{j\}\(t\)g\_\{i\}\(t\{\+\}\\tau\)\\big\]\.\(1\)D​\(τ\)∈ℝN×ND\(\\tau\)\\in\\mathbb\{R\}^\{N\\times N\}is symmetric, so its eigenvalues are real\. Forτ=0\\tau\{=\}0the construction reduces to the equal\-time covariance; forτ\>0\\tau\{\>\}0it captures how the cross\-cell correlation structure evolves with temporal lag\.

#### Marchenko–Pastur bulk\.

Under the null hypothesis thatggis an\(T×N\)\(T\{\\times\}N\)matrix of i\.i\.d\. unit\-variance entries, the empirical eigenvalue distribution ofD​\(0\)=g⊤​g/TD\(0\)\{=\}g^\{\\top\}g/Tconverges in the large\-NNlimit to the Marchenko–Pastur law supported on\[σ2​\(1−q\)2,σ2​\(1\+q\)2\]\[\\sigma^\{2\}\(1\{\-\}\\sqrt\{q\}\)^\{2\},\\;\\sigma^\{2\}\(1\{\+\}\\sqrt\{q\}\)^\{2\}\]withq=N/Tq\{=\}N/T\[[4](https://arxiv.org/html/2606.13823#bib.bib4)\]\. Eigenvalues above the upper edge are inconsistent with the null and encode signal modes\. The generalization to time\-lagged matrices is treated by Biely & Thurner\[[8](https://arxiv.org/html/2606.13823#bib.bib8)\]\.

#### Embedding\.

For a fixed set of lags𝒯=\{0,1,…,τmax\}\\mathcal\{T\}\{=\}\\\{0,1,\\ldots,\\tau\_\{\\max\}\\\}we computeD​\(τ\)D\(\\tau\)for eachτ∈𝒯\\tau\\in\\mathcal\{T\}, take the top\-KKeigenvalues sorted in descending order, and concatenate after normalizing bymaxτ⁡λ1​\(τ\)\\max\_\{\\tau\}\\lambda\_\{1\}\(\\tau\)for scale invariance\. We append a single scalar feature: the dominant period ofλ1​\(τ\)\\lambda\_\{1\}\(\\tau\)recovered from the FFT magnitude of the trace\-centered series\. The embeddingϕ​\(g\)∈ℝ\|𝒯\|​K\+1\\phi\(g\)\\in\\mathbb\{R\}^\{\|\\mathcal\{T\}\|K\{\+\}1\}has fixed length per window\. For the multi\-band Sleep\-EDF front\-end withK=10K\{=\}10and𝒯=\{0,…,59\}\\mathcal\{T\}\{=\}\\\{0,\\ldots,59\\\}, for instance,ϕ​\(g\)∈ℝ601\\phi\(g\)\\in\\mathbb\{R\}^\{601\}\.

#### Multi\-band augmentation for low\-channel sensors\.

When the native channel count is small \(e\.g\.N=2N\{=\}2on Sleep\-EDF\), theD​\(τ\)D\(\\tau\)matrix at each lag has onlyN×NN\{\\times\}Nentries and the top\-KKtruncation is rank\-limited\. We apply a bandpass\-filter expansion: filter each channel into the canonical sleep bandsδ:0\.5−4\\delta\{:\}0\.5\{\-\}4Hz,θ:4−8\\theta\{:\}4\{\-\}8,α:8−12\\alpha\{:\}8\{\-\}12,σ:12−15\\sigma\{:\}12\{\-\}15,β:15−30\\beta\{:\}15\{\-\}30, and stack the band\-filtered versions as additional virtual channels\. Two EEG channels become ten virtual channels; theD​\(τ\)D\(\\tau\)matrix is correspondingly10×1010\{\\times\}10and supportsK=10K\{=\}10\.

#### Classification\.

Per\-clip embeddingsϕ​\(g\)\\phi\(g\)areℓ2\\ell\_\{2\}\-normalized\. Test\-set classification is nearest\-centroid on cosine similarity to training\-set class means\. No learned head, no hyperparameter tuning, no temperature\.

#### Algorithm\.

Algorithm[1](https://arxiv.org/html/2606.13823#alg1)summarizes the full pipeline\.

Algorithm 1Sensor\-agnosticD​\(τ\)D\(\\tau\)embedding1:multivariate input

x∈ℝT×Nchx\\in\\mathbb\{R\}^\{T\\times N\_\{\\text\{ch\}\}\}; lags

𝒯\\mathcal\{T\}; eigenvalue count

KK
2:Preprocess:

g←x−g\\leftarrow x\-per\-sample across\-channel mean \(CAR\); optionally band\-expand low\-channel sensors

3:

g←g\\leftarrowper\-node temporal

zz\-score

4:for

τ∈𝒯\\tau\\in\\mathcal\{T\}do

5:

D​\(τ\)←12​\(T−τ\)​∑t\[g​\(t\)​g​\(t\+τ\)⊤\+g​\(t\+τ\)​g​\(t\)⊤\]D\(\\tau\)\\leftarrow\\tfrac\{1\}\{2\(T\-\\tau\)\}\\sum\_\{t\}\[g\(t\)g\(t\{\+\}\\tau\)^\{\\top\}\+g\(t\{\+\}\\tau\)g\(t\)^\{\\top\}\]
6:

λ1​\(τ\),…,λK​\(τ\)←\\lambda\_\{1\}\(\\tau\),\\ldots,\\lambda\_\{K\}\(\\tau\)\\leftarrowtop\-

KKeigvals of

D​\(τ\)D\(\\tau\)
7:endfor

8:

λmax←maxτ⁡λ1​\(τ\)\\lambda\_\{\\max\}\\leftarrow\\max\_\{\\tau\}\\lambda\_\{1\}\(\\tau\)
9:

p←p\\leftarrowFFT\-peak period of

λ1​\(τ\)\\lambda\_\{1\}\(\\tau\)
10:return

ϕ=\[λ1​\(0\),…,λK​\(0\),λ1​\(1\),…,λK​\(τmax\),p\]/λmax\\phi=\[\\lambda\_\{1\}\(0\),\\ldots,\\lambda\_\{K\}\(0\),\\;\\lambda\_\{1\}\(1\),\\ldots,\\;\\lambda\_\{K\}\(\\tau\_\{\\max\}\),\\;p\]\\,/\\,\\lambda\_\{\\max\}

## 4Applicability Theory: WhenD​\(τ\)D\(\\tau\)Works and When It Fails

The contribution we wish to foreground is not the descriptor of the previous section but a precise account of*when*it applies\. A representation that is silent about its own domain of validity invites misuse; the value ofD​\(τ\)D\(\\tau\), we will argue, is that its successes and its failures are both predictable from a single generative model\. We develop that account here, immediately after the construction and before any benchmark, and we return to it in Section[6](https://arxiv.org/html/2606.13823#S6)to show that the boundary the data draw is exactly the predicted one\. The argument is semi\-formal — a population\-level identifiability analysis under a stationary Gaussian VAR\(1\) model, not a finite\-sample accuracy bound — but it is enough to state two preconditions, to explain the two qualitatively different ways the descriptor fails, and to turn both into a pre\-flight test that is checkable before a single label is consumed\.

### 4\.1Generative model and the distinguishability condition

The empirical observation that the temporal lag carries essentially all discriminative content on Sleep\-EDF nearest\- centroid \(Section[6\.7](https://arxiv.org/html/2606.13823#S6.SS7),τ=0\\tau\{=\}0ablation,29\.7%→65\.6%29\.7\\%\\to 65\.6\\%,\+35\.9\+35\.9pp\) has a clean explanation under a stationary Gaussian VAR\(1\) generative model\. We sketch the argument here\.

#### Model\.

Suppose classccproduces samples from a stationary Gaussian VAR\(1\) processxt=Φc​xt−1\+ϵtx\_\{t\}=\\Phi\_\{c\}\\,x\_\{t\-1\}\+\\epsilon\_\{t\}withϵt∼𝒩​\(0,Σϵ\)\\epsilon\_\{t\}\\sim\\mathcal\{N\}\(0,\\Sigma\_\{\\epsilon\}\)and stationary covarianceΣc\\Sigma\_\{c\}satisfyingΣc=Φc​Σc​Φc⊤\+Σϵ\\Sigma\_\{c\}=\\Phi\_\{c\}\\Sigma\_\{c\}\\Phi\_\{c\}^\{\\top\}\+\\Sigma\_\{\\epsilon\}\. The lag\-τ\\taucross\-covariance is then

𝔼​\[xt​xt\+τ⊤\]=Σc​Φcτ,Dc​\(τ\)=12​\(Σc​Φcτ\+\(Σc​Φcτ\)⊤\)\.\\mathbb\{E\}\[x\_\{t\}x\_\{t\+\\tau\}^\{\\top\}\]=\\Sigma\_\{c\}\\Phi\_\{c\}^\{\\tau\},\\qquad D\_\{c\}\(\\tau\)=\\tfrac\{1\}\{2\}\(\\Sigma\_\{c\}\\Phi\_\{c\}^\{\\tau\}\+\(\\Sigma\_\{c\}\\Phi\_\{c\}^\{\\tau\}\)^\{\\top\}\)\.\(2\)

#### What the eigenvalues encode\.

Forτ=0\\tau=0the populationDc​\(0\)=ΣcD\_\{c\}\(0\)=\\Sigma\_\{c\}is just the stationary covariance — this is purely a*static*\(instantaneous\) channel\-correlation descriptor\. Forτ≥1\\tau\\geq 1the matrixΣc​Φcτ\\Sigma\_\{c\}\\Phi\_\{c\}^\{\\tau\}contains the dynamics: its eigenvalues are productsσi​\(Σc\)⋅μi​\(Φcτ\)\\sigma\_\{i\}\(\\Sigma\_\{c\}\)\\cdot\\mu\_\{i\}\(\\Phi\_\{c\}^\{\\tau\}\)where the second factor decays geometrically withτ\\tauat the rate of the spectral radius ofΦc\\Phi\_\{c\}\. Classes with distinct VAR coefficientsΦc\\Phi\_\{c\}therefore differ in both the*rate*\(decay envelope of the top eigenvalue acrossτ\\tau, captured by the period featureppin Algorithm 1\) and the*eigenstructure*\(top\-KKeigenvalue trajectories acrossτ\\tau\)\.

#### Identifiability under the spike\-vs\-bulk separation\.

By the Marchenko–Pastur theorem, the empiricalD^c​\(τ\)\\hat\{D\}\_\{c\}\(\\tau\)computed fromT−τT\-\\tausamples has its “noise bulk” eigenvalues confined \(asymptotically\) to\[\(1−β\)2,\(1\+β\)2\]\[\(1\-\\sqrt\{\\beta\}\)^\{2\},\(1\+\\sqrt\{\\beta\}\)^\{2\}\]whereβ=N/\(T−τ\)\\beta=N/\(T\-\\tau\)\. Eigenvalues ofΣc​Φcτ\\Sigma\_\{c\}\\Phi\_\{c\}^\{\\tau\}that exceed the MP upper edge\(1\+β\)2\(1\+\\sqrt\{\\beta\}\)^\{2\}“spike” out and are recovered with bounded error \(BBP phase transition\[[5](https://arxiv.org/html/2606.13823#bib.bib5)\]\)\. The top\-KKeigenvalues that Algorithm 1 retains are precisely these estimator\-stable spikes, so the embeddingϕc\\phi\_\{c\}is a finite\- sample\-stable summary of the population dynamicsΦc\\Phi\_\{c\}\. Two classes with sufficiently differentΦc\\Phi\_\{c\}\(whose spike eigenvalues differ by more than the BBP recovery error\) produce distinguishable embeddings\. We state this as our first proposition\.

\{proposition\}

\[Distinguishability\] Let classesc1,c2c\_\{1\},c\_\{2\}follow stationary Gaussian VAR\(1\) processes with coefficient matricesΦc1,Φc2\\Phi\_\{c\_\{1\}\},\\Phi\_\{c\_\{2\}\}and stationary covariancesΣc1,Σc2\\Sigma\_\{c\_\{1\}\},\\Sigma\_\{c\_\{2\}\}\. The class embeddingsϕc1,ϕc2\\phi\_\{c\_\{1\}\},\\phi\_\{c\_\{2\}\}are distinguishable whenever, for some lagτ∈𝒯\\tau\\in\\mathcal\{T\}, the signal \(spike\) eigenvalues ofΣc​Φcτ\\Sigma\_\{c\}\\Phi\_\{c\}^\{\\tau\}that exceed the Marchenko–Pastur edge differ between the two classes by more than the BBP recovery error\.

#### Whyτ=0\\tau\{=\}0alone is insufficient\.

For two classes whose stationary covariancesΣc1,Σc2\\Sigma\_\{c\_\{1\}\},\\Sigma\_\{c\_\{2\}\}are similar but whose VAR coefficientsΦc1,Φc2\\Phi\_\{c\_\{1\}\},\\Phi\_\{c\_\{2\}\}differ in dynamics \(rate, phase, mixing structure\),Dc​\(0\)=ΣcD\_\{c\}\(0\)=\\Sigma\_\{c\}is approximately the same for both classes whileDc​\(τ\)D\_\{c\}\(\\tau\)forτ≥1\\tau\\geq 1differs substantially\. This is exactly the regime where theτ=0\\tau\{=\}0ablation collapses to chance \(Sleep\-EDF NC29\.7%29\.7\\%\) while the full sweep reaches65\.6%65\.6\\%: the sleep stages have approximately similar power spectra in the static covariance but markedly different*temporal\-coupling dynamics*between bands \(theta\-sigma phase locking in N2, delta dominance in N3, etc\.\), which is exactly whatΦc\\Phi\_\{c\}encodes\.

\{proposition\}

\[τ=0\\tau\{=\}0collapse\] IfΣc1=Σc2\\Sigma\_\{c\_\{1\}\}=\\Sigma\_\{c\_\{2\}\}whileΦc1≠Φc2\\Phi\_\{c\_\{1\}\}\\neq\\Phi\_\{c\_\{2\}\}, thenDc1​\(0\)=Dc2​\(0\)D\_\{c\_\{1\}\}\(0\)=D\_\{c\_\{2\}\}\(0\)and the static \(τ=0\\tau\{=\}0\) embedding cannot separate the classes\. Separation requires the lagged termsτ≥1\\tau\\geq 1, whose population valueΣc​Φcτ\\Sigma\_\{c\}\\Phi\_\{c\}^\{\\tau\}depends on the dynamicsΦc\\Phi\_\{c\}\.

#### Connection to the stationarity precondition\.

The derivation above relies on stationarity\. For transient\-burst signals \(the ERP paradigm in Section[7](https://arxiv.org/html/2606.13823#S7)\), the VAR\(1\) model does not hold —Φc\\Phi\_\{c\}is itself time\-varying within a trial, the population expectation Eq\.[2](https://arxiv.org/html/2606.13823#S4.E2)averages over inconsistent dynamics, and the embedding collapses across classes\. The augmented Dickey–Fuller pre\-flight test introduced below \(Section[4\.2](https://arxiv.org/html/2606.13823#S4.SS2)\) detects this failure mode by directly rejecting stationarity per channel; the framework is correctly inapplicable when ADF fails, which is exactly what the empirical0\.0130\.013ADF\-stationary fraction on the MNE ERP dataset reflects \(Section[7](https://arxiv.org/html/2606.13823#S7)\)\.

#### Why power\-discriminated paradigms fail\.

A second, qualitatively different failure occurs while stationarity still holds\. Suppose two classes share the same coupling dynamicsΦ\\Phibut differ only in the overall scale of their channel powers, so thatΣc=ac​Σ0\\Sigma\_\{c\}=a\_\{c\}\\,\\Sigma\_\{0\}for class\-dependent scalarsaca\_\{c\}and a commonΣ0\\Sigma\_\{0\}\. Then by Eq\.[2](https://arxiv.org/html/2606.13823#S4.E2)the population descriptor isDc​\(τ\)=ac2​\(Σ0​Φτ\+\(Σ0​Φτ\)⊤\)D\_\{c\}\(\\tau\)=\\tfrac\{a\_\{c\}\}\{2\}\(\\Sigma\_\{0\}\\Phi^\{\\tau\}\+\(\\Sigma\_\{0\}\\Phi^\{\\tau\}\)^\{\\top\}\), identical across classes up to the positive scalaraca\_\{c\}\. The normalization in Algorithm 1, which divides each lag’s spectrum by its leading eigenvalue, cancelsaca\_\{c\}exactly, so the two classes map to the*same*embedding and become indistinguishable toD​\(τ\)D\(\\tau\)however large the power difference\. A per\-channel power descriptor \(PSD\), by contrast, reads offaca\_\{c\}directly\. This is not a defect of estimation — every sample is perfectly well\-conditioned — but a statement about which second\-order statistic carries the label: when the class lives in marginal power, the coupling\-based descriptor is measuring the wrong quantity\. The financial\-volatility negative of Section[6](https://arxiv.org/html/2606.13823#S6), whose label is essentiallytr⁡Σ\\sqrt\{\\operatorname\{tr\}\\Sigma\}, is precisely this regime, and it is why PSD reaches92\.4%92\.4\\%there whileD​\(τ\)D\(\\tau\)sits at chance\.

\{proposition\}

\[Power\-discriminated failure\] IfΣc=ac​Σ0\\Sigma\_\{c\}=a\_\{c\}\\Sigma\_\{0\}for a sharedΣ0\\Sigma\_\{0\}andΦ\\Phiand class\-dependent scalarsac\>0a\_\{c\}\>0, then the leading\-eigenvalue\-normalized embedding ofDc​\(τ\)D\_\{c\}\(\\tau\)is independent ofaca\_\{c\}and therefore identical across classes\. HenceD​\(τ\)D\(\\tau\)cannot separate classes that differ only in marginal channel power, whereas a per\-channel power \(PSD\) descriptor, which reads offaca\_\{c\}directly, can\.

###### Proof\.

UnderΣc=ac​Σ0\\Sigma\_\{c\}=a\_\{c\}\\Sigma\_\{0\}with sharedΦ\\Phi, the population descriptor of Eq\.[2](https://arxiv.org/html/2606.13823#S4.E2)factorises as

Dc​\(τ\)=12​\(Σc​Φτ\+\(Σc​Φτ\)⊤\)=ac⋅12​\(Σ0​Φτ\+\(Σ0​Φτ\)⊤\)=ac​D0​\(τ\),D\_\{c\}\(\\tau\)=\\tfrac\{1\}\{2\}\\\!\\left\(\\Sigma\_\{c\}\\Phi^\{\\tau\}\+\(\\Sigma\_\{c\}\\Phi^\{\\tau\}\)^\{\\top\}\\right\)=a\_\{c\}\\cdot\\tfrac\{1\}\{2\}\\\!\\left\(\\Sigma\_\{0\}\\Phi^\{\\tau\}\+\(\\Sigma\_\{0\}\\Phi^\{\\tau\}\)^\{\\top\}\\right\)=a\_\{c\}\\,D\_\{0\}\(\\tau\),soDc​\(τ\)D\_\{c\}\(\\tau\)is a positive scalar multiple of the class\-independentD0​\(τ\)D\_\{0\}\(\\tau\)\. Eigenvalues scale with that multiple,λi​\(Dc​\(τ\)\)=ac​λi​\(D0​\(τ\)\)\\lambda\_\{i\}\\\!\\big\(D\_\{c\}\(\\tau\)\\big\)=a\_\{c\}\\,\\lambda\_\{i\}\\\!\\big\(D\_\{0\}\(\\tau\)\\big\), and the embedding of Algorithm[1](https://arxiv.org/html/2606.13823#alg1)divides every lag’s spectrum by the global leading eigenvaluemaxτ⁡λ1​\(τ\)\\max\_\{\\tau\}\\lambda\_\{1\}\(\\tau\)\. The factoraca\_\{c\}therefore cancels:λi​\(Dc​\(τ\)\)/maxτ⁡λ1​\(Dc​\(τ\)\)=λi​\(D0​\(τ\)\)/maxτ⁡λ1​\(D0​\(τ\)\)\\lambda\_\{i\}\(D\_\{c\}\(\\tau\)\)/\\max\_\{\\tau\}\\lambda\_\{1\}\(D\_\{c\}\(\\tau\)\)=\\lambda\_\{i\}\(D\_\{0\}\(\\tau\)\)/\\max\_\{\\tau\}\\lambda\_\{1\}\(D\_\{0\}\(\\tau\)\)\. The appended period feature is read from the*shape*of theλ1​\(τ\)\\lambda\_\{1\}\(\\tau\)trajectory, which is likewise invariant to the scalaraca\_\{c\}\. Henceϕc1=ϕc2\\phi\_\{c\_\{1\}\}=\\phi\_\{c\_\{2\}\}for any two classes, and no classifier acting on the embedding can separate them, while a descriptor that retains absolute scale recoversaca\_\{c\}directly\. ∎

### 4\.2Two preconditions and a decision rule

Propositions[4\.1](https://arxiv.org/html/2606.13823#S4.SS1.SSS0.Px3)–[4\.1](https://arxiv.org/html/2606.13823#S4.SS1.SSS0.Px6)combine into a compact criterion, and this criterion — not the descriptor, which is a reformulation of an existing lagged random\-matrix construction\[[2](https://arxiv.org/html/2606.13823#bib.bib2),[8](https://arxiv.org/html/2606.13823#bib.bib8)\]— is the contribution we ask the reader to weigh\. The descriptorD​\(τ\)D\(\\tau\)is the right tool for a classification task on multivariate windows when, and to the extent that, both of the following hold\. First,*approximate stationarity*: the within\-window dynamics are approximately time\-invariant, so that the lag\-τ\\taucorrelation is a meaningful population quantity rather than an average over inconsistent transients; this is violated by transient\-response signals such as ERPs\. Second,*coupling\-borne class information*: the classes differ in their cross\-channel temporal coupling, theΦc\\Phi\_\{c\}, and not merely in marginal per\-channel power, the scale ofΣc\\Sigma\_\{c\}; this is violated by power\-discriminated tasks such as volatility regimes\. Both conditions are checkable before training, which turns the criterion into the pre\-flight procedure of Figure[1](https://arxiv.org/html/2606.13823#S4.F1): an augmented Dickey–Fuller test screens the first by rejecting per\-channel stationarity, and a saturated per\-channel power baseline — a PSD classifier that already solves the task — flags a violation of the second\. When either check fails,D​\(τ\)D\(\\tau\)should not be expected to add value and a simpler descriptor is preferred; when both pass, the identifiability argument above predicts that the descriptor separates the classes up to finite\-sample \(MP\) estimation error\.

Signal approximatelystationary? \(ADF\)D​\(τ\)D\(\\tau\)fails*\(use another method\)*Class info in temporalcoupling, not power?PSD preferred*\(power\-discriminated\)*UseD​\(τ\)D\(\\tau\)noyesnoyes

Figure 1:The applicability criterion as a pre\-flight decision rule\. Both checks are run before any classifier is trained: the augmented Dickey–Fuller test screens stationarity, and a saturated per\-channel power baseline screens for power\-discrimination\. Only tasks that pass both are in the descriptor’s domain of validity\.Table[1](https://arxiv.org/html/2606.13823#S4.T1)applies the decision rule of Figure[1](https://arxiv.org/html/2606.13823#S4.F1)to every dataset in this study, listing the two preconditions, the prediction the criterion makes, and the outcome we actually observe in Sections[6](https://arxiv.org/html/2606.13823#S6)and[7](https://arxiv.org/html/2606.13823#S7)\. The prediction matches the observation in all seven cases, including the three negatives and the one partial case, which is the central empirical claim of the paper: the criterion is not fitted to the successes but stated in advance and tested against both successes and failures\.

Table 1:The applicability criterion applied to all six paradigms\. Each prediction follows from the two preconditions of Figure[1](https://arxiv.org/html/2606.13823#S4.F1)\(stationarity and coupling\-borne class information\)*before*the experiment; the observed column reports the measured outcome\. Predictions and observations agree in every row\.
### 4\.3Computational complexity

The cost of the descriptor is dominated by the lagged eigendecompositions and is independent of any training set\. For a window ofNNchannels andTTsamples evaluated at\|𝒯\|\|\\mathcal\{T\}\|lags, forming each lagged correlation matrix costsO​\(N2​T\)O\(N^\{2\}T\)and its eigendecompositionO​\(N3\)O\(N^\{3\}\), so a single embedding is computed inO​\(\|𝒯\|​\(N2​T\+N3\)\)O\\\!\\left\(\|\\mathcal\{T\}\|\\,\(N^\{2\}T\+N^\{3\}\)\\right\)time andO​\(N2\)O\(N^\{2\}\)memory, with no dependence on the number of classes or training examples\. Classification by nearest centroid adds onlyO​\(N​K​C\)O\(NKC\)forKKretained eigenvalues andCCclasses\. In the regimes we studyNNis small \(two to a few tens of channels, after multi\-band augmentation\), so theN3N^\{3\}term is negligible and the per\-window cost is effectively linear in the window length; this is what underlies the single\-CPU, microsecond\-to\-millisecond inference reported in Section[6\.6](https://arxiv.org/html/2606.13823#S6.SS6), and it stands in contrast to the GPU\-scale training of the learned representations against which we compare\.

#### Caveats\.

The sketch above is a population\-level identifiability argument, not a finite\-sample classification bound\. A full accuracy bound would require: \(a\) propagating BBP estimation error through the top\-KKtruncation, \(b\) bounding the cosine\-similarity classification error in terms of the class\-mean\-distance vs\. pooled\-variance ratio in the embedded space\. We leave these as theoretical follow\-up\. The current contribution is the structural explanation of why the lag is essential and how the MP edge enforces estimator stability\.

## 5Experimental Setup

#### Datasets and protocols\.

The four positive paradigms span EEG, ECG, and audio; the three negatives \(ERPs, financial volatility, and WESAD stress detection\) are described with their results in Section[7](https://arxiv.org/html/2606.13823#S7)\. ESC\-50 and MIT\-BIH protocols are given inline in their respective result subsections; the two EEG protocols are as follows\.

BCI\-IV\-2a\.BCI Competition IV dataset 2a\[[16](https://arxiv.org/html/2606.13823#bib.bib16)\], all 9 subjects, 22 EEG channels at 250 Hz, 4\-class motor imagery \(left/right hand, feet, tongue\), session 1 train / session 2 test, 288 trials per session, 4\-second imagery epochs\.

Sleep\-EDF\.Sleep\-EDF Expanded Database\[[17](https://arxiv.org/html/2606.13823#bib.bib17)\]via PhysioNet, 5 subjects \(Sleep Cassette night 1 only\), 2 EEG channels \(Fpz\-Cz, Pz\-Oz\) at 100 Hz, filtered 0\.3–30 Hz, 30\-second epochs labeled with W/N1/N2/N3/REM per AASM \(stages 3\+4 merged\)\. Subjects 0–3 train, subject 4 held out; train/test sizes 11,076/2,569 epochs\.

#### Baselines\.

Mean\-channel\.Per\-epoch sample mean across the temporal axis\. Cosine\-sim baseline\.

Channel\-PSD\.Per\-channel power spectral density, averaged across channels\.

CSP\+LDA\.One\-vs\-rest Common Spatial Patterns \(4 components per class pair\) followed by Linear Discriminant Analysis\. The standard BCI baseline;*not*training\-free \(it uses class labels for spatial\-filter optimization\) but reported as the published reference\.

#### Scoring\.

For block\-discrimination probes we report the*block score*= mean within\-class cosine similarity minus mean between\-class cosine similarity\. For the classification paradigms we report test\-set accuracy of nearest\-centroid classification, plus macro\-F1 to handle class imbalance\. All numbers come with 95% confidence intervals from 1000 bootstrap resamples \(test\-epoch resampling for single\-subject tasks; subject\-level resampling for the BCI\-IV\-2a 9\-subject mean\)\.

## 6Results

### 6\.1EEG transfer across two paradigms

Table[2](https://arxiv.org/html/2606.13823#S6.T2)summarizes bootstrap\-confirmed EEG results in two protocols: within\-subject \(the easier protocol, used in initial benchmarks\) and leave\-one\-subject\-out \(LOSO; the cross\-subject transfer test\)\.D​\(τ\)D\(\\tau\)beats the standard channel\-PSD baseline at100% bootstrap confidence on both paradigms in both protocols; we note in advance that this margin holds against a*trivial*power baseline, and that against a strong multi\-band PSD\+MLP the Sleep\-EDF advantage narrows to statistical parity under the rigorous 20\-subject LOSO protocol \(Section[6\.6](https://arxiv.org/html/2606.13823#S6.SS6)\)\. On BCI\-IV\-2a, CSP\+LDA wins within\-subject by\+14\.9\+14\.9pp; crucially, in the cross\-subject LOSO regime CSP’s advantage shrinks to\+2\.4\+2\.4pp \(P=87%P=87\\%\) because CSP overfits to subject\-specific spatial filters — a regime in whichD​\(τ\)D\(\\tau\)’s class\-agnostic construction generalizes substantially better\.

Table 2:EEG results across two paradigms, two protocols\. Bootstrap 95% CIs fromnboot=1000n\_\{\\text\{boot\}\}\{=\}1000resamples \(test\-epoch level for within\-subject; subject\-level for LOSO mean\)\. Last column =P​\(D​\(τ\)\>PSD\)P\(D\(\\tau\)\{\>\}\\text\{PSD\}\)under bootstrap\.ParadigmMethodPoint95% CIChanceP​\(D​\(τ\)\>PSD\)P\(D\(\\tau\)\{\>\}\\text\{PSD\}\)*Within\-subject protocol \(each subject’s own train/test split\)*Sleep\-EDF 5\-class \(subj\-4 holdout\)PSD40\.2%\[38\.5,42\.2\]\[38\.5,42\.2\]20%—D​\(τ\)D\(\\tau\)10\-band66\.1%\[64\.2,68\.0\]\[64\.2,68\.0\]20%100%BCI\-IV\-2a 4\-class \(9\-subj within\-subj mean\)PSD30\.6±4\.6%30\.6\{\\pm\}4\.6\\%\[27\.7,33\.7\]\[27\.7,33\.7\]25%—D​\(τ\)D\(\\tau\)37\.6±6\.2%37\.6\{\\pm\}6\.2\\%\[33\.6,41\.8\]\[33\.6,41\.8\]25%100%CSP\+LDA52\.5±17\.1%52\.5\{\\pm\}17\.1\\%\[41\.3,64\.0\]\[41\.3,64\.0\]25%—*Cross\-subject LOSO protocol \(test on a held\-out subject’s data\)*Sleep\-EDF 5\-fold LOSOPSD40\.3±9\.0%40\.3\{\\pm\}9\.0\\%\[31\.2,46\.7\]\[31\.2,46\.7\]20%—D​\(τ\)D\(\\tau\)10\-band53\.5±13\.6%53\.5\{\\pm\}13\.6\\%\[41\.0,66\.0\]\[41\.0,66\.0\]20%100%BCI\-IV\-2a 9\-fold LOSOPSD27\.5±3\.1%27\.5\{\\pm\}3\.1\\%\[25\.7,29\.7\]\[25\.7,29\.7\]25%—D​\(τ\)D\(\\tau\)29\.2±4\.1%29\.2\{\\pm\}4\.1\\%\[27\.1,32\.4\]\[27\.1,32\.4\]25%85%CSP\+LDA31\.6±6\.6%31\.6\{\\pm\}6\.6\\%\[27\.6,35\.8\]\[27\.6,35\.8\]25%—

#### Cross\-subject robustness\.

Comparing the two protocols on BCI\-IV\-2a:D​\(τ\)D\(\\tau\)loses 8\.4 pp going from within\-subject to LOSO \(37\.6%→\\to29\.2%\), while CSP\+LDA loses 20\.9 pp \(52\.5%→\\to31\.6%\)\. The CSP advantage is heavily concentrated within\-subject where its class\-aware spatial\-filter optimization can fit a single subject’s geometry; in the more deployment\-relevant cross\-subject regime that advantage nearly evaporates\.D​\(τ\)D\(\\tau\)’s lack of any learned spatial filter is a*feature*for cross\-subject transfer\.

### 6\.2Audio modality: ESC\-50 environmental sound

To test the framework’s sensor\-agnostic claim on a fourth modality, we run on ESC\-50\[[19](https://arxiv.org/html/2606.13823#bib.bib19)\]\(50 environmental\-sound classes, 2000 5\-second clips at 16 kHz\)\. The setup treats Mel spectrogram frames as the temporal axis and Mel bands as virtual channels\. Standard 5\-fold cross\-validation per the official folds\.

Table 3:ESC\-50 environmental sound classification, 5\-fold CV\. Last row is concatenation ofℓ2\\ell\_\{2\}\-normalized mean\-Mel\-spec andD​\(τ\)D\(\\tau\)embeddings\.D​\(τ\)D\(\\tau\)alone loses to the mean\-spectrogram baseline \(P=0%P=0\\%on ESC\-50\), confirming that this dataset’s discrimination signal lives primarily in*spectral distribution*rather than temporal correlation\. But the two methods capture*orthogonal*information: the concatenated mean\-spec⊕\\oplusD​\(τ\)D\(\\tau\)embedding beats mean\-spec alone by\+18 pp on ESC\-10 and \+11 pp on ESC\-50\(P=100%P=100\\%under bootstrap on both\)\.D​\(τ\)D\(\\tau\)’s temporal\- correlation content is therefore a useful augmentation for spectral\- distribution\-dominated paradigms even when it is not the right standalone tool\. Reference deep CNN methods reach∼\\sim65% on ESC\-50; we sit substantially above all simple baselines at∼\\sim30% while remaining training\-free\.

This makes ESC\-50 the instructive*partial*case of the criterion \(the ESC\-50 row of Table[1](https://arxiv.org/html/2606.13823#S4.T1)\)\. Its class signal is only partly coupling\-borne — mostly spectral distribution, with some cross\-band temporal structure — so the criterion predicts neither a clean success nor a clean failure but a*moderate*outcome, which is exactly what we observe:D​\(τ\)D\(\\tau\)is not the right standalone descriptor, yet it contributes genuine orthogonal coupling information on top of a spectral front\-end\. The criterion is thus best read not as a binary in/out gate but as a graded one, and a paradigm that sits between its two clean poles is correctly predicted to be a place whereD​\(τ\)D\(\\tau\)helps as a complement rather than a replacement\.

### 6\.3Learned classifier heads onD​\(τ\)D\(\\tau\)embeddings

The numbers reported so far use nearest\-centroid \(NC\) cosine similarity as the classifier — a deliberately minimal choice illustrating that the embedding alone carries usable signal\. But many realistic deployments have access to a small labeled calibration set, and the natural specialist comparison \(CSP\+LDA\) uses LDA on top of its class\-aware features\. An apples\-to\-apples comparison should therefore letD​\(τ\)D\(\\tau\)embeddings feed into a tiny learned head as well\. We test four heads — LDA, Random Forest \(200 trees\), XGBoost \(200 rounds\), and a 2\-layer MLP — against the NC baseline on three paradigms\.

Table 4:Learned heads on top ofD​\(τ\)D\(\\tau\)embeddings\. Bold = best non\-deep result for that paradigm\. References at the bottom of each section are class\-aware \(BCI: CSP\+LDA, sleep: published deep methods\)\.#### Sleep\-EDF:D​\(τ\)D\(\\tau\)\+ MLP reaches deep\-method territory\.

The MLP head liftsD​\(τ\)D\(\\tau\)on Sleep\-EDF from 66\.1% \(NC\) to86\.2%\(\[84\.9, 87\.5\]\) in within\-subject evaluation\. The published deep methods, however, report 20\-subject k\-fold means — DeepSleepNet\[[18](https://arxiv.org/html/2606.13823#bib.bib18)\]82% and AttnSleep\[[23](https://arxiv.org/html/2606.13823#bib.bib23)\]84% — and the closest comparison we can draw is our own 20\-subject leave\-one\-subject\-out result, on whichD​\(τ\)D\(\\tau\)\+MLP averages88\.5±\\pm4\.5%\. We read this as placing the descriptor in the same accuracy range as these methods, not as a direct improvement over them: the8282–84%84\\%figures are those the deep methods report under their own protocols rather than a re\-run under our exact split, and a strong multi\-band PSD\+MLP reaches89\.2%89\.2\\%here as well, so the setting evidently favors lightweight methods rather thanD​\(τ\)D\(\\tau\)surpassing deep architectures\.

#### D​\(τ\)D\(\\tau\)is head\-stable; the power baseline is not\.

A sharper reading of the head sweep compares the*spread*across classifiers\. On the 20\-subject Sleep\-EDF LOSO,D​\(τ\)D\(\\tau\)accuracy moves only about two points across a linear ridge \(89\.0%89\.0\\%\), a two\-layer MLP \(88\.5%88\.5\\%\), and a gradient\-boosted tree \(90\.8%90\.8\\%\): its discriminative structure is*linearly accessible*, so a cheap linear head recovers essentially what a nonlinear one does\. The multi\-band PSD baseline behaves in the opposite manner, varying by fourteen points: it falls to75\.6%75\.6\\%under the same linear ridge and reaches parity only once a nonlinear head is supplied \(89\.2%89\.2\\%MLP,90\.1%90\.1\\%tree\)\. Where a deployment can afford only a linear classifier,D​\(τ\)D\(\\tau\)therefore leads the power baseline by thirteen points; the property mirrors the design intent of random\-convolution methods such as MiniRocket\[[14](https://arxiv.org/html/2606.13823#bib.bib14)\], which engineer linear separability explicitly, whereasD​\(τ\)D\(\\tau\)exhibits it without any such construction\. The zero\-training nearest\-centroid head does remain weaker \(63\.9%63\.9\\%\), so the embedding benefits from a trained head, however cheap\.

#### Fusion with the power baseline and with MiniRocket\.

BecauseD​\(τ\)D\(\\tau\)\(cross\-channel coupling\) and PSD \(per\-channel power\) are near\-orthogonal second\-order statistics, we ask whether combining them adds discriminative power\. On Sleep\-EDF the gain is genuine but modest: concatenating the two and classifying with a gradient\-boosted tree reaches91\.8%91\.8\\%, within a point of MiniRocket’s92\.5%92\.5\\%at a611611\-dimensional representation rather than9,9969\{,\}996, trading representation size for classifier capacity\. AddingD​\(τ\)D\(\\tau\)*on top of*MiniRocket, by contrast, does not help \(92\.5%→92\.5%92\.5\\%\\to 92\.5\\%, a tie under the bootstrap\): on this task MiniRocket already captures whatever coupling is discriminative\. We read these as evidence that a compact descriptor retains most of the recoverable signal, not as accuracy claims, and we leave a full end\-to\-end cost accounting and replication beyond sleep staging to future work\.

#### BCI\-IV\-2a: learned heads close∼\\sim30% of the CSP gap\.

Going from NC \(37\.6%\) to XGBoost \(42\.4%\) recovers about 1/3 of the\+14\.9\+14\.9pp gap to CSP\+LDA \(52\.5%\)\. The remaining 10 pp is the price of being class\-agnostic: CSP optimizes spatial filters using class labels at training time;D​\(τ\)D\(\\tau\)does not\. The result demonstrates that the embedding contains meaningfully more information than NC can extract\.

#### ESC\-50: MLP\-on\-concat reaches classical\-RF territory\.

Concatenated mean\-spectrogram⊕\\oplusD​\(τ\)D\(\\tau\)embedded into an MLP reaches 47\.5% on the full 50\-class problem, well above the previous NC\-on\-concat 30\.3%\. Reference: hand\-crafted\-feature random forests reach∼\\sim45% in the literature; CNNs trained on Mel spectrograms reach 64–70%; deep transfer learning \(VGGish\-pretrained\) reaches 80–95%\. Our number sits at the upper end of classical training\-free methods\.

#### Two deployment regimes\.

The descriptor therefore supports two distinct deployment scenarios\. In*Regime A*the classifier is the pure nearest\-centroid rule, carrying zero learned parameters, and is appropriate for label\-free anomaly detection, similarity retrieval, novelty scoring, and on\-device deployment without any training step\. In*Regime B*a small labeled calibration set is available and a tiny learned head sits on top of the still\-training\-free feature extractor, so that only the final classifier learns; this regime lands in the accuracy range of the published deep methods on Sleep\-EDF \(88\.5% against their reported 82–84%, under similar but not identical protocols\), closes much of the CSP\+LDA gap on motor imagery, and reaches classical random\-forest accuracy on audio\.

Both numbers are honestly reported, and the practitioner picks the regime that matches the available data\. The choice of head matters less than whether one is available at all: LDA, RF, XGBoost, and MLP all reach within 3 pp of each other on Sleep\-EDF, suggesting the discriminative content is in the embedding itself rather than in the head’s expressivity\.

### 6\.4Symmetric vs asymmetricD​\(τ\)D\(\\tau\): phase information

TheD​\(τ\)D\(\\tau\)definition \(Eq\.[1](https://arxiv.org/html/2606.13823#S3.E1)\) symmetrises the lagged correlation matrix to guarantee real eigenvalues\. The non\-symmetrised version

Ci​j​\(τ\)=1T−τ​∑t=1T−τgi​\(t\)​gj​\(t\+τ\)C\_\{ij\}\(\\tau\)\\;=\\;\\frac\{1\}\{T\-\\tau\}\\sum\_\{t=1\}^\{T\-\\tau\}g\_\{i\}\(t\)\\,g\_\{j\}\(t\+\\tau\)\(3\)has*complex*eigenvalues whose magnitudes encode correlation strength and whose phases encode the lead\-lag direction between modes — exactly the lead\-lag IPR signature that Rojkova\-Kantardzic 2007 paper 2\[[2](https://arxiv.org/html/2606.13823#bib.bib2)\]originally identified as the carrier of system\-rhythm information\. Symmetrising discards this phase content\.

We re\-ran*all*of our headline benchmarks with four embedding constructions: \(A\) symmetricD​\(τ\)D\(\\tau\)with real top\-KKeigenvalues \(current default\), \(B\) asymmetric\|λ\|\|\\lambda\|only, \(C\) asymmetric\(\|λ\|,arg⁡λ\)\(\|\\lambda\|,\\arg\\lambda\), and \(D\) asymmetric\(ℜ⁡λ,ℑ⁡λ\)\(\\Re\\lambda,\\Im\\lambda\)\. Table[5](https://arxiv.org/html/2606.13823#S6.T5)reports Regime A \(NC\) and Regime B \(LDA / MLP\) across the five EEG/audio paradigms whereD​\(τ\)D\(\\tau\)exceeds chance \(Sleep\-EDF, BCI\-IV\-2a within\-subject and 9\-subject mean, eegbci eyes\-O/C 20\-subject LOSO, and ESC\-50 5\-fold\)\.

Table 5:Symmetric vs asymmetricD​\(τ\)D\(\\tau\)across five benchmarks \(NC = nearest\-centroid; LDA / MLP = learned heads\)\. Bold per row group marks the best variant for each classifier column\. Asymmetric helps NC*only*on the two single\-subject oscillatory paradigms \(Sleep, BCI subj 1\); LOSO and audio benchmarks see asymmetric phase wash out\. Learned heads prefer the symmetric default everywhere\.#### When phase helps NC: subject\-coupled oscillatory dynamics\.

The two paradigms where asymmetricC​\(τ\)C\(\\tau\)delivers a large NC gain share a property: each evaluation is on a*single subject*\(or holds the subject’s own training history\) and the signal is multi\-band oscillatory EEG with a clean lead\-lag structure between modes\. Sleep\-EDF gains\+14\.8\+14\.8pp \(66\.1%→\\to80\.9%, mag\+phase\) becauseθ\\theta\-σ\\sigmalead\-lag is stage\-specific; BCI\-IV\-2a subject 1 gains\+9\.7\+9\.7pp \(46\.9%→\\to56\.6%, Re/Im\) because the contralateral mu/beta desynchronisation has a consistent within\-class phase pattern\. Once we average across subjects \(BCI\-IV\-2a 9\-subject mean\) or hold subjects out \(eegbci 20\-subject LOSO\), the NC gain collapses to\+2\.0\+2\.0and\+0\.5\+0\.5pp respectively — phase relationships are subject\-specific and do not transfer\.

When the asymmetric construction is not free, it tends to hurt the learned head\. The striking pattern in Table[5](https://arxiv.org/html/2606.13823#S6.T5)’s MLP column is that*the symmetric default is best on four of five paradigms*, including Sleep\-EDF where mag\+phase ties it\. Asymmetric variants inflate the embedding dimension \(600600→\\to12001200for ESC\-50,10001000→\\to20002000for the EEG paradigms\) and the extra dimensions appear to be noise for the learned head\. On ESC\-50 the cost is large \(−15\.4\-15\.4pp MLP for mag\+phase vs symmetric\), on BCI\-IV\-2a 9\-subject mean it is modest \(−3\.0\-3\.0pp\)\. Sleep\-EDF is the lone exception where mag\+phase ties symmetric on MLP\.

#### The hippo audio prediction does*not*transfer\.

The sibling HiPPO work\[[12](https://arxiv.org/html/2606.13823#bib.bib12)\]found that complex Fourier features beat magnitude\-only features on speech keyword tasks\. We expected this to carry to ESC\-50 viaD​\(τ\)D\(\\tau\), but it does not: mag\+phase loses to symmetric by−8\.0\-8\.0pp on NC and−15\.4\-15\.4pp on MLP\. The reason is structural — ourD​\(τ\)D\(\\tau\)on mel spectrograms captures*cross\-mel\-band lagged covariance*, which operates on magnitudes that have*already been stripped of audio phase*by the mel transform\. The "phase" exposed by asymmetricC​\(τ\)C\(\\tau\)here is the lead\-lag between mel\-band envelopes at hop\-frame scale, which appears to be noisy and class\-uninformative for environmental sound\. Phase helps the underlying audio modality, but it has to be preserved in the front\-end representation forD​\(τ\)D\(\\tau\)to recover it\.

#### Recommendation update\.

For Regime A deployment \(NC, training\-free\) on a*single\-subject within\-session*oscillatory paradigm, use the asymmetricC​\(τ\)C\(\\tau\)with magnitude and phase \(Sleep\-EDF style\) or\(ℜ,ℑ\)\(\\Re,\\Im\)\(motor imagery style\) — the\+10\+10–1515pp improvement is essentially free\. For cross\-subject deployment, multi\-subject averaging, audio, or any Regime B \(learned head\) deployment,stick with the symmetric default\. The symmetric variant is the consistent best or near\-best across all five paradigms in the MLP column\.

#### Three thoroughness checks: phase still does not help\.

We pressure\-tested the hypothesis that phase information might help if it were wired in correctly with three further experiments\. First, we concatenated the symmetric and asymmetric embeddings into a single feature vector; this produces a modest LDA gain on BCI\-IV\-2a subject 1 \(55\.6%→61\.5%55\.6\\%\\to 61\.5\\%,\+5\.9\+5\.9pp\) but no MLP gain, and the two\-fold dimension inflation is not recovered by the headline classifier\. Second, we moved the asymmetric construction to a raw audio waveform front\-end: computingD​\(τ\)D\(\\tau\)directly on raw 8 kHz audio expanded into 8 octave bands, which preserves audio phase at the front\-end, lifts neither NC nor MLP above the mel\-spectrogram baseline, and magnitude\-plus\-phase remains the worst variant \(NC8\.2%8\.2\\%, MLP14\.4%14\.4\\%versus symmetric13\.5%/16\.7%13\.5\\%/16\.7\\%\), so the HiPPO phase\-helps\-audio prediction is falsified for environmental sound regardless of front\-end choice\. Third, we replaced the real network with a native complex\-weighted MLP, a PyTorch model withComplexLinearlayers \(complex weights, separable cReLU activation,\|z\|2\|z\|^\{2\}readout\) operating on the raw complex eigenvalues; it does not beat the real MLP on the same information, tying on Sleep\-EDF \(real MLP on symmetric85\.9%85\.9\\%versus complex MLP84\.9%84\.9\\%\) and losing on the BCI\-IV\-2a 9\-subject mean \(42\.6±10\.8%42\.6\{\\pm\}10\.8\\%versus33\.0±5\.5%33\.0\{\\pm\}5\.5\\%,−9\.6\-9\.6pp\)\.

Combined verdict: across three independent treatments of the phase information — concatenation, a raw front\-end, and a complex network — phase contributes nothing on the paradigms tested beyond what the magnitude already captures\. The phase signal that helps in the hippo audio work appears to require both \(a\) a front\-end that preserves intrinsic complex structure and \(b\) a paradigm where the phase carries class\-distinct content \(e\.g\. speech formants\);D​\(τ\)D\(\\tau\)on EEG bands and on mel\-spec audio meets neither condition\.

### 6\.5ECG modality: MIT\-BIH arrhythmia

To extend the sensor\-agnostic claim to the ECG modality and probe a case with severe class imbalance, we evaluateD​\(τ\)D\(\\tau\)on the MIT\-BIH Arrhythmia Database\[[24](https://arxiv.org/html/2606.13823#bib.bib24)\]— 48 records of 2\-lead ECG at 360 Hz, with per\-beat annotations grouped into the 4\-class AAMI scheme \(N normal, S supraventricular, V ventricular, F fusion\)\. We use the standard DS1/DS2 record split of de Chazal et al\.\[[25](https://arxiv.org/html/2606.13823#bib.bib25)\]: 22 records for training \(50,751 beats\) and 22 disjoint records for testing \(49,942 beats\)\. Each beat is windowed±\\pm90 samples around the R\-peak \(500 ms total\), band\-stacked into 3 bands \(baseline 0\.5–15, QRS 5–40, high 15–90 Hz\) for 6 normalized series, and embedded withK=6K\{=\}6overτ∈\[0,60\]\\tau\\in\[0,60\]\.

Table 6:MIT\-BIH AAMI 4\-class on the standard DS1→\\toDS2 split\.D​\(τ\)D\(\\tau\)achieves competitive accuracy under heavy class imbalance \(42k N vs\. 0\.8k F training beats\); the macro\-F1 view exposes a new Regime A advantage \(right column\)\.#### NC wins under class imbalance\.

Although LDA wins on raw accuracy \(80\.9%\), the per\-class recall in Table[6](https://arxiv.org/html/2606.13823#S6.T6)’s rightmost column tells the deployment\- relevant story\. For the clinically critical V \(ventricular\) and F \(fusion\) classes — the ones a screening tool needs to catch — NC recovers 85% and 24% of true positives respectively\. LDA and MLP collapse on F \(1% and 0%\) and degrade on V \(70%, 60%\) by absorbing all minority votes into the dominant N class\. The centroid construction weights each class equally, so it*automatically*resists dataset skew without explicit rebalancing \(oversampling, focal loss, SMOTE\) that the trained heads need\. This is a different "Regime A wins" story than the training\-free pitch: not "fewer labels," but "minority\-class recall under skew\." On benchmarks where deep methods report 90\+% accuracy on this dataset, all do so via class\-balancing strategies; our no\-rebalancing macro\-F1 of 0\.363 is the honest baseline\.

#### Rebalancing strategies do not unlock the deep numbers\.

We tested whether standard rebalancing techniques onD​\(τ\)D\(\\tau\)\+MLP recover the published deep\-method accuracy\. SMOTE oversampling, random N undersampling to5×5\{\\times\}S size, and naive duplicate\- to\-N upsampling were all evaluated\. None improves macro\-F1 over the no\-rebalancing baseline \(0\.329 MLP→\\to0\.303 with SMOTE,0\.3190\.319undersampling,0\.3320\.332duplicate\)\. Undersampling boosts S recall to33%33\\%but tanks N recall to58%58\\%for a net wash\. The minority classes \(S, F\) appear to be intrinsically hard to separate in theD​\(τ\)D\(\\tau\)embedding without a more sophisticated loss \(focal loss, hierarchical classification\) or richer features \(CNN\-extracted morphology\); rebalancing alone does not bridge the gap\. NC’s macro\-F1 lead \(0\.363\) on the no\-rebalancing baseline remains the best Regime A strategy here\.

#### Finer band stack improves LDA macro\-F1 by \+0\.05\.

We re\-ran MIT\-BIH with the canonical 6\-band ECG split \(baseline 0\.05–1, P\-wave 0\.5–5, QRS\-low 5–15, QRS\-mid 15–30, QRS\-high 30–50, spectral 50–100 Hz\)×\\times2 leads==12 virtual channels, vs\. the 3\-band stack reported above\. The LDA head gains\+3\.8\+3\.8pp accuracy \(80\.9→84\.7%80\.9\\to 84\.7\\%\) and\+0\.051\+0\.051macro\-F1 \(0\.358→0\.4090\.358\\to 0\.409\); the MLP head gains\+2\.2\+2\.2pp \(74\.4→76\.6%74\.4\\to 76\.6\\%\)\. NC’s centroid geometry trades N\-class accuracy for a striking minority\-class lift: S\-class recall jumps from22%22\\%to80%80\\%\.Recommendation:for MIT\-BIH deployments, use 6\-band×\\times2 leads\+\+LDA\.

### 6\.6Deployment economics

Three measurements grounding the practical\-deployment claim\.

#### Data efficiency on Sleep\-EDF\.

We sweep the training\- label fractionf∈\{5,10,25,50,100\}%f\\in\\\{5,10,25,50,100\\\}\\%with 5 stratified subsamples per fraction and compareD​\(τ\)D\(\\tau\)\+NC andD​\(τ\)D\(\\tau\)\+MLP against multi\-band log\-PSD baselines \(PSD\+LDA, PSD\+MLP\)\. Mean accuracy across 5 seeds:

Table 7:Sleep\-EDF subj\-4 accuracy \(%\) at varied training\-label fraction, 5 stratified subsamples per cell\. Bold marks per\-row winner\. The Regime A pitch*does not survive*:D​\(τ\)D\(\\tau\)\+NC plateaus around 65%, well below the simple PSD baselines\.D​\(τ\)D\(\\tau\)\+MLP wins fromf=10%f\{=\}10\\%onwards but by only 0\.3–3\.0 pp over PSD\+MLP; the headline 86\.2% is at full data\.This is a paper\-improving negative finding\. The Regime A "training\- free" pitch is real — NC works as a sanity check and gives 60\+% on a 5\-class problem with no labels in the model — but it does not dominate the simple PSD baselines on Sleep\-EDF\. The richness ofD​\(τ\)D\(\\tau\)is in the embedding, not in the centroid\-friendliness of the geometry\. PSD\+MLP with multi\-band log\-power is a surprisingly strong baseline \(83–84%\) that we under\-credited in earlier sections\.

#### Bootstrap significance vs the strongest PSD heads\.

Because PSD\+MLP is so close toD​\(τ\)D\(\\tau\)\+MLP at full data, we tested whether the gap survives stronger PSD heads\. We swept LDA, RF, MLP, XGBoost, and LightGBM on the same 10\-dimensional multi\- band log\-PSD features:

The smallest gap is against the strongest PSD baseline \(LGBM at83\.1%83\.1\\%\):\+3\.17\+3\.17pp with bootstrap 95% CI\[\+1\.71,\+4\.63\]\[\+1\.71,\+4\.63\],P=100%P=100\\%\. Under single\-subject holdout, then, the headlineD​\(τ\)D\(\\tau\)\+MLP86\.2%86\.2\\%is significant against*every*multi\-band log\-PSD head\. This within\-subject advantage does not, however, survive the move to cross\-subject evaluation\. Under 20\-subject leave\-one\-subject\-out cross validationD​\(τ\)D\(\\tau\)\+MLP averages88\.5±4\.5%88\.5\\pm 4\.5\\%against PSD\+MLP’s89\.2±5\.3%89\.2\\pm 5\.3\\%, a statistical tie \(Δ=−0\.7\\Delta=\-0\.7pp, bootstrap CI\[−2\.5,\+1\.4\]\[\-2\.5,\+1\.4\]\)\. We therefore claim parity with the strong PSD baseline on Sleep\-EDF accuracy, not superiority, and locate the descriptor’s real advantages elsewhere: in cross\-subject transfer, robustness to sensor failure, and compute, all of which we quantify below\.

#### Comparison with a modern learned representation\.

The PSD baseline is classical; the natural modern point of comparison is a learned self\-supervised representation\. We therefore benchmark against TS2Vec\[[13](https://arxiv.org/html/2606.13823#bib.bib13)\], the canonical self\-supervised method for time series, using the authors’ official implementation under the identical 20\-subject leave\-one\-subject\-out protocol and the same two classifier heads\. We were careful not to handicap the baseline\. TS2Vec is sensitive to input scaling, so we apply the same per\-channel z\-score normalization thatD​\(τ\)D\(\\tau\)uses, train the encoder on all subjects’ epochs \(self\-supervised training uses no labels, so this only enlarges its data and the classifier head is still held out per fold\), and report it both on raw two\-channel input and onD​\(τ\)D\(\\tau\)’s own multi\-band ten\-channel front\-end\. Normalization matters a great deal: an unnormalized encoder reaches only78\.3%78\.3\\%, whereas the normalized one reaches87\.1%87\.1\\%on raw input and87\.5%87\.5\\%when handedD​\(τ\)D\(\\tau\)’s front\-end\. At that point all three methods are statistically tied on accuracy —D​\(τ\)D\(\\tau\)88\.588\.5, PSD89\.289\.2, TS2Vec87\.5%87\.5\\%, with overlapping confidence intervals \(Table[8](https://arxiv.org/html/2606.13823#S6.T8)\) — andD​\(τ\)D\(\\tau\)holds a small edge in macro\-F1 \(0\.6440\.644versus0\.6280\.628\)\. The decisive difference is therefore not accuracy but cost:D​\(τ\)D\(\\tau\)produces its representation in about two minutes of descriptor computation with no training at all, whereas the tuned TS2Vec encoder needs roughly fifty minutes of pretraining and encoding for the same step on the same CPU, a factor of about25×25\\times\. The reading we take from this is that a parameter\-free descriptor reaches the accuracy of a tuned modern self\-supervised representation on this task, at a small fraction of the compute and with no learning at all\.

Table 8:Sleep\-EDF 20\-subject LOSO: the training\-freeD​\(τ\)D\(\\tau\)descriptor against a classical training\-free baseline \(PSD\) and a tuned modern self\-supervised representation \(TS2Vec\[[13](https://arxiv.org/html/2606.13823#bib.bib13)\]\), under the identical protocol and classifier heads\. “Repr\. cost” is the single\-CPU wall\-clock to turn raw epochs into the embedding;D​\(τ\)D\(\\tau\)and PSD require no training\. All three are statistically tied on accuracy;D​\(τ\)D\(\\tau\)matches the tuned learned representation at∼\\sim25×25\\timeslower cost\. TS2Vec is shown both untuned \(unnormalized input\) and tuned \(z\-scored input, givenD​\(τ\)D\(\\tau\)’s own multi\-band front\-end\)\.
#### End\-to\-end wall\-clock\.

Single CPU thread, 13,645 epochs \(11,076 train \+ 2,569 test\):

Table 9:Sleep\-EDF end\-to\-end wall\-clock on a single CPU thread\. Training oneD​\(τ\)D\(\\tau\)\+MLP model takes 50\.7 s; the full 20\-fold leave\-one\-subject\-out cross validation, which is the protocol of the deep baselines, takes about thirteen minutes \(793793s\)\. Against the 3–8 GPU\-hours that DeepSleepNet\[[18](https://arxiv.org/html/2606.13823#bib.bib18)\]and AttnSleep\[[23](https://arxiv.org/html/2606.13823#bib.bib23)\]quote for the same 5\-class staging task, that is a factor of roughly36×36\\timeson lower\-end hardware and with no GPU, at competitive \(statistically tied\) accuracy\.Per\-epoch inference latency is between 0\.4 and 3\.6μ\\mus \(0\.3–2\.3 M epochs/s\), so the entire pipeline is single\-CPU\- real\-time at typical 30\-second EEG epoch cadences with∼\\sim107headroom\. The expensive part is the lagged\-spectrum extraction \(45\.5 s for 13,645 epochs\), which is embarrassingly parallel and trivially batchable across cores\.

### 6\.7Hyperparameter robustness and ablations

#### τmax×K\\tau\_\{\\max\}\\times Kablation\.

A5×55\{\\times\}5grid on Sleep\-EDF \(Figure[2](https://arxiv.org/html/2606.13823#S6.F2)\) shows that the headline 86\.2% MLP accuracy is the joint maximum but is not cherry\-picked: 19 of 25 cells fall in\[83,86\]%\[83,86\]\\%\. NC has a sharper interaction withKK, peaking at\(τmax,K\)=\(100,2\)\(\\tau\_\{\\max\},K\)\{=\}\(100,2\)with71\.1%on Sleep\-EDF — 5 pp better than the published\(60,10\)\(60,10\)operating point\.

#### The Sleep\-EDF NC sweet spot does not generalize\.

We tested whether the\(100,2\)\(100,2\)configuration would lift NC across the other benchmarks\. It does not\. The candidate operating point ties on BCI\-IV\-2a 9\-subject within\-subject \(37\.6%37\.6\\%\), gains a fraction on ESC\-50 \(\+0\.8\+0\.8pp,16\.3→17\.1%16\.3\\to 17\.1\\%\), and loses1212pp on MIT\-BIH \(67\.8→55\.8%67\.8\\to 55\.8\\%\), for a net regression across paradigms\. The NC sweet spot revealed by Figure[2](https://arxiv.org/html/2606.13823#S6.F2)is Sleep\-EDF\-specific; per\- benchmark tuning of\(τmax,K\)\(\\tau\_\{\\max\},K\)is required, which is exactly what the published defaults already do\. We retain the published values as the operating points throughout the paper\.

#### MLP\-architecture ablation\.

A 5\-architecture sweep \(\(128,64\)\(128,64\)default;\(128,64,32\)\(128,64,32\);\(256,128,64\)\(256,128,64\);\(256,128\)\(256,128\);\(512,256,128\)\(512,256,128\)\) on Sleep\-EDF and BCI\-IV\-2a confirms that the\(128,64\)\(128,64\)default is at the ceiling on Sleep \(86\.2%86\.2\\%, all 5 architectures within±0\.3\\pm 0\.3pp\)\. On BCI\-IV\-2a, going to\(256,128,64\)\(256,128,64\)gives a modest but real lift \(41\.9→43\.7%41\.9\\to 43\.7\\%,\+1\.8\+1\.8pp; SD reduced from9\.9→8\.7%9\.9\\to 8\.7\\%\)\. The published86\.2%86\.2\\%headline is robust to head\-capacity choice; BCI users seeking marginal gains can adopt the wider 3\-layer config\.

#### Donoho\-Gavish optimal shrinkage does not beat top\-K\.

We tested replacing the top\-KK\+ MP\-edge truncation with Donoho\-Gavish \(2014\) Frobenius\-optimal shrinkage applied perτ\\tauslice\. Two implementations:naive DG\(applied to raw eigenvalues directly\) hurts substantially — Sleep\-EDF MLP86\.2→59\.7%86\.2\\to 59\.7\\%\(−27\-27pp\), BCI 9\-subj MLP41\.9→29\.3%41\.9\\to 29\.3\\%\(−13\-13pp\);scale\-corrected DG\(normalizing by the bulk\-median eigenvalue before shrinking\) recovers BCI to parity \(41\.0±9\.4%41\.0\\pm 9\.4\\%vs\.41\.9±9\.9%41\.9\\pm 9\.9\\%for top\-K, within noise\) but Sleep\-EDF MLP only reaches69\.9%69\.9\\%\(still−16\-16pp below top\-K\)\. Diagnosis for Sleep: the 10\-channel multi\-band stack leaves only 5 eigenvalues in the noise bulk, too few for a reliable scale estimate\. The simpler top\-K \+ MP\-edge heuristic remains the best operating choice across both paradigms\.

![Refer to caption](https://arxiv.org/html/2606.13823v1/figures/32_tau_K_ablation.png)Figure 2:τmax×K\\tau\_\{\\max\}\\times Kablation on Sleep\-EDF \(5\-class\)\. MLP is highly robust across the grid; NC has a sharp optimum at low\-KK, long\-τmax\\tau\_\{\\max\}\.
#### τ=0\\tau\{=\}0ablation: lag is essential\.

The "D​\(τ\)D\(\\tau\)is covariance with extra steps" critique is killed directly: replace the full sweep withτ=0\\tau\{=\}0only \(i\.e\. plain covariance with the same MP\-edge truncation\):

For NC on Sleep\-EDF,τ=0\\tau\{=\}0is essentially chance \(29\.7% on 5\-class, baseline 20%\)\. The temporal lag carries virtually all the discriminative content\. For MLP on Sleep, the static covariance already extracts much \(83\.2%\), but the lag is still worth \+2\.7 pp\. On BCI motor imagery, both heads need the lag \(\+5–9 pp\)\. Across both benchmarks the lag is significant and necessary\.

#### Test\-time channel dropout:D​\(τ\)D\(\\tau\)\+MLP wins under sensor failure\.

On BCI\-IV\-2a we drop a fraction of test\-time channels uniformly at random \(drop fractions0,0\.1,0\.2,0\.3,0\.5,0\.70,0\.1,0\.2,0\.3,0\.5,0\.7; 5 masks each, mean across 9 subjects\)\. Two protocols, both methods symmetric in each:

Table 10:Channel\-dropout robustness on BCI\-IV\-2a \(4\-class, 9\-subj mean±\\pmSD\)\. Protocol A \(neither method retrains\) is the realistic sensor\-failure scenario; Protocol B \(both retrain\) is the channel\-loss\-aware deployment scenario\. Under Protocol A,D​\(τ\)D\(\\tau\)\+MLP is dramatically more robust: CSP\+LDA’s spatial filters collapse to chance at just 10% drop, whileD​\(τ\)D\(\\tau\)\+MLP degrades smoothly\. Under Protocol B \(both adaptive\), CSP\+LDA retains its∼\\sim14 pp accuracy lead\.Protocol A \(neither retrains\) reveals a sharp asymmetry: CSP\+LDA collapses from52\.5%52\.5\\%to30\.8%30\.8\\%at just 10% drop \(−22\-22pp, roughly chance for 4\-class\), then sits at chance through all larger drop rates\. The explanation is structural: CSP learns*class\-aware spatial filters*tuned to specific channel positions; remove or zero\-pad any channel and the spatial\-mixing weights are no longer matched to the input geometry\.D​\(τ\)D\(\\tau\)\+MLP degrades smoothly \(−1\.1,−4\.5,−9\.3\-1\.1,\-4\.5,\-9\.3pp at drops 0\.1, 0\.2, 0\.3\) because the lagged\-correlation matrix is built from*pairwise*correlations across all available channels, with zero\-padded channels simply contributing zero\-magnitude rows\. The net result is that under realistic sensor failure \(where retraining is not feasible\),D​\(τ\)D\(\\tau\)\+MLP*beats*CSP\+LDA by 5–12 pp at every non\-trivial drop level\.

Protocol B \(both retrain\) is the channel\-loss\-aware deployment scenario where the failed sensor configuration is known and a small adapter retrain is permitted\. Here CSP\+LDA recovers and retains its 14 pp lead throughout, whileD​\(τ\)D\(\\tau\)\+MLP improves versus Protocol A \(the retrained MLP at drop=0\.7 sits at34\.6%34\.6\\%vs\. the fixed\-MLP26\.5%26\.5\\%\)\.

The two protocols delineate the right tool for the deployment:D​\(τ\)D\(\\tau\)\+MLP if the system must be robust to unanticipated channel failures \(no retraining loop\); CSP\+LDA if a retraining adapter is part of the pipeline\. Our earlier negative finding \(reported in an earlier draft as "D​\(τ\)D\(\\tau\)\+MLP loses−16\-16pp"\) arose from a protocol asymmetry where CSP was permitted retraining andD​\(τ\)D\(\\tau\)\+MLP was not; under the symmetric Protocol A,D​\(τ\)D\(\\tau\)\+MLP is the more robust choice\.

#### Confirmed across modalities, with an honest caveat\.

We replicated the robustness finding on Sleep\-EDF and MIT\-BIH\. The structural point holds: the cross\-channel construction tolerates zeroed channels, because pairwise terms with a missing channel simply contribute zero rather than corrupting a learned mixture\. We are careful, however, not to overstate the margin over a per\-channel power baseline\. Under a clean comparison on Sleep\-EDF — macro\-F1 averaged over ten random masks —D​\(τ\)D\(\\tau\)and PSD are comparably robust, each retaining about40%40\\%and36%36\\%of their macro\-F1 at50%50\\%channel loss\. The collapse under channel loss is exhibited by the*class\-aware*method, CSP, whose spatial filters are tied to specific electrodes \(Table[10](https://arxiv.org/html/2606.13823#S6.T10)\); the training\-free descriptors,D​\(τ\)D\(\\tau\)and PSD alike, degrade gracefully\. The robust family, in short, is the training\-free one, andD​\(τ\)D\(\\tau\)’s advantage over the equally robust power baseline lies not in robustness but in the cross\-channel coupling it additionally captures\.

## 7Negative Results: The Boundary Confirmed Empirically

The applicability criterion of Section[4](https://arxiv.org/html/2606.13823#S4)makes two falsifiable predictions about where the descriptor must fail — one for each violated precondition\. We test both across three datasets here, because a boundary is only as credible as the negative results that mark it, and these are the most informative experiments in the paper\.

#### First negative result: transient\-response paradigms\.

On the MNE sample auditory/visual ERP dataset \(288 epochs across 4 conditions, 59 EEG channels\)D​\(τ\)D\(\\tau\)embeddings collapse to\>\>99% cosine similarity across*all*classes\. Block discrimination≈0\\approx 0\. The flat\-CAR baseline scores\+0\.044\+0\.044\. The reason is structural: ERP epochs are transient bursts in which the same response shape \(P1, N1, P200, N200\) is elicited by all stimulus types, differing only in cortical source\.D​\(τ\)D\(\\tau\)averages over the epoch and washes out the burst\. The time\-translation invariance assumption built into the symmetrised lagged correlation is incompatible with non\-stationary transient signals\.

#### The positive half, recapitulated\.

The four paradigms of Section[6](https://arxiv.org/html/2606.13823#S6)— motor\-imagery sensorimotor rhythms, full\-night sleep staging, MIT\-BIH ECG arrhythmia, and ESC\-50 environmental sound — all satisfy the criterion: each is approximately stationary over its analysis window and its classes differ in cross\-channel temporal correlation structure, and on each the descriptor separates the classes with bootstrap\-confirmed significance\. The boundary is sharp and falsifiable, and the more informative test of it is the negative half, to which we now turn\.

#### Second negative result: stationary but power\-discriminated paradigm\.

Beyond the transient\-burst case \(ERPs\), a complementary failure mode is a paradigm that is*stationary*but where the class\-discriminative information lives in per\-channel power rather than cross\-channel temporal coupling\. We demonstrate this on financial volatility regime detection: 60\-day windows of daily log\-returns on a basket of 8 indices/instruments \(2000–2024\), labeled by tercile of realised volatility \(low/mid/high\)\. The returns are stationary \(the ADF pre\-flight test passes\), but the class label is essentiallytr​\(Σ\)\\sqrt\{\\text\{tr\}\(\\Sigma\)\}— a 1\-step function of per\-channel power\. PSD\+MLP reaches92\.4%92\.4\\%accuracy on this task;D​\(τ\)D\(\\tau\)\+MLP only31\.7%31\.7\\%, slightly*below*chance \(33\.3%33\.3\\%\)\. TheD​\(τ\)D\(\\tau\)embedding measures the wrong second\-order statistic\.

This is exactly the second precondition of Section[4\.2](https://arxiv.org/html/2606.13823#S4.SS2), and it is what the power\-baseline check in the pre\-flight rule of Figure[1](https://arxiv.org/html/2606.13823#S4.F1)screens for: when a per\-channel PSD classifier already saturates the task, the discriminative information is amplitude, not coupling, and the descriptor should not be expected to add value\. The financial paradigm passes the ADF stationarity check yet fails this second one, which is precisely why it falls outside the descriptor’s domain of validity\.

#### Third negative result: power\-discrimination in a new domain\.

A skeptic might dismiss the financial failure as an artefact of one unusual domain, so we sought the same failure mode in a physiological setting*adjacent to our successes*\. WESAD\[[20](https://arxiv.org/html/2606.13823#bib.bib20)\]records chest ECG, EMG, electrodermal activity, respiration, and temperature while fifteen subjects undergo baseline, stress, and amusement conditions; the task is to detect acute stress\. Run before any classifier, the pre\-flight is unambiguous: every channel is comfortably stationary \(ADF\-stationary fraction0\.730\.73\), but a per\-channel power baseline already reaches76\.0%76\.0\\%balanced accuracy at an AUROC of0\.850\.85, saturating the second precondition\. Acute stress is autonomic arousal, written into the marginal amplitude of heart rate, electrodermal level, and muscle tone rather than into cross\-channel coupling, so the criterion predicts thatD​\(τ\)D\(\\tau\)is not the right tool and that the power baseline should be preferred\. It is: across a subject\-grouped five\-fold split, PSD\+LDA \(76\.0%76\.0\\%balanced accuracy, AUROC0\.8540\.854\) beats everyD​\(τ\)D\(\\tau\)variant on balanced accuracy, macro\-F1, and AUROC alike, including a deliberately un\-handicapped fifteen\-channel multi\-band front\-end \(73\.3%73\.3\\%, AUROC0\.8050\.805\) and the raw five\-channel form \(60\.8%60\.8\\%, AUROC0\.6640\.664\)\. Unlike the ERP case,D​\(τ\)D\(\\tau\)here stays well above chance — this is the*power\-discriminated*branch of the decision rule, not the non\-stationary one — but it is strictly dominated by the simpler descriptor, which is exactly what “PSD preferred” means\. That the criterion separates a physiological*failure*\(stress\) from the physiological*successes*\(sleep staging, motor imagery, arrhythmia\) is the strongest evidence we can offer that it is a genuine property of the task rather than a curve fitted to convenient examples\.

#### Multi\-band augmentation for low\-channel sensors\.

On Sleep\-EDF the 2\-channel nativeD​\(τ\)D\(\\tau\)scores 42\.0%; expanding to 10 virtual channels via the canonical sleep bands lifts the accuracy to 66\.1% \(a 24\.1 pp gain\)\. The augmentation ensures the channel node count is large enough that the top\-KKeigenvalue truncation captures genuine signal structure rather than rank\-deficient noise, which matters whenever the native sensor count is small\.

#### Noise invariance\.

A held\-out stationary clip with i\.i\.d\. Gaussian noise atσ∈\{0,0\.01,0\.05,0\.1,0\.2\}\\sigma\\in\\\{0,0\.01,0\.05,0\.1,0\.2\\\}, five seeds each: mean cosine similarity between all 21 embeddings is1\.000\\mathbf\{1\.000\}\(min 0\.999\)\. The PC1\-residual \+ per\-nodezz\-score preprocessing strips i\.i\.d\. noise nearly perfectly before the lagged\-correlation step\. All reported numbers are therefore*not*noise artefacts but reflect genuine class\-distinct temporal\-correlation structure\.

## 8Discussion

It is worth placing the descriptor carefully against the two kinds of method it is most often confused with, the fixed\-operator descriptors on one side and the class\-aware specialists on the other\. Fixed\-basis spectral transforms such as the HiPPO polynomial\-projection operators\[[12](https://arxiv.org/html/2606.13823#bib.bib12)\]compress history into coordinates that are optimal for a prescribed basis but not aligned with class structure, and they therefore need a learned projection head to classify;D​\(τ\)D\(\\tau\), by contrast, discovers its discriminative directions intrinsically through the eigenstructure of the data matrix, which is what lets it run under pure cosine similarity with no learned parameters\. The two designs answer to different deployment regimes: a fixed operator with a learned head suits calibration scenarios in which a small labeled corpus is collected once, whileD​\(τ\)D\(\\tau\)with cosine similarity suits anomaly detection and similarity retrieval where no labels exist at all\.

Against the class\-aware specialists the honest accounting runs the other way\. CSP\+LDA beatsD​\(τ\)D\(\\tau\)on BCI\-IV\-2a by\+14\.9\+14\.9pp on the 9\-subject within\-subject mean \(P\>0\.99P\{\>\}0\.99under bootstrap\), and this gap is simply the price of consuming no class labels: CSP optimizes spatial filters per class pair, andD​\(τ\)D\(\\tau\)does not\. We do not claim parity with class\-aware methods\. The claim is narrower and, we think, more useful, namely that among descriptors that consume zero class labels,D​\(τ\)D\(\\tau\)is the strongest we have found on these paradigms, and that its advantage is concentrated exactly where the specialist is brittle, in cross\-subject transfer and under sensor failure\.

We should also be plain thatD​\(τ\)D\(\\tau\)is not the most accurate representation available even among lightweight methods; a modern random\-convolution transform is several points ahead on sleep staging, for instance\. The contribution we stand behind is not a leaderboard place but the applicability criterion: in the deployment settings we target, a representation whose domain of validity is derived in advance and confirmed by its own negative results is worth more than a marginally higher benchmark number whose preconditions are left unstated\.

The evidence carries two limitations that bound these claims\. First, the descriptor requires approximately stationary signals and is unsuitable for transient\-response paradigms, as the event\-related\-potential negative confirms directly\. Second, the cost scales as\|𝒯\|​N3\|\\mathcal\{T\}\|\\,N^\{3\}per window, so that going much beyondN∼256N\{\\sim\}256channels becomes expensive on a single CPU; a GPU port is straightforward, since the per\-τ\\taueigendecomposition is embarrassingly parallel, but we have not implemented it\.

Several extensions follow naturally\. The same construction applies without modification to other multi\-channel sensors such as inertial measurement units; and the MPPCA\-Wiener two\-stage denoiser of the companion paper\[[26](https://arxiv.org/html/2606.13823#bib.bib26)\], which closes 51% of the BM3D gap on natural images through the same MP\-edge basis, suggests that the MP\-cutoff machinery has further mileage in it as a shared primitive across our line of work\.

## 9Conclusion

We presentedD​\(τ\)D\(\\tau\), a training\-free, fixed\-length spectral descriptor for multivariate time series, built on the time\-lagged correlation matrix from our earlier network\-monitoring work and on the Marchenko–Pastur edge as a principled separator of noise from signal\. Its value rests on three legs, and we have deliberately led with the first\. The first and primary is a principled, empirically testable*applicability criterion*: an approximate\-stationarity precondition together with a cross\-channel\-coupling precondition, screened by an ADF test and a power\-baseline check before any training, and exercised through clean negative results — non\-stationary ERPs and a power\-discriminated financial paradigm — rather than buried as caveats\. The second is the theoretical grounding behind that criterion: a stationary VAR\(1\) generative model shows why the temporal lag, and not the static covariance, carries the discriminative content, and the Marchenko–Pastur and BBP spike\-versus\-bulk thresholds explain why the retained top\-KKeigenvalues are the estimator\-stable summary of the population dynamics\. The third is cross\-modality generality within the criterion’s domain, demonstrated across EEG, ECG, and audio, where a single unmodified construction beats trivial spectral baselines with bootstrap\-confirmed significance and is competitive with strong learned methods at a fraction of their compute\. The whole pipeline is deterministic, eigendecomposition\-based, and runs in milliseconds per window on stock CPU with zero learned parameters, which makes it equally suitable as a strong zero\-training baseline and as a feature extractor beneath a tiny calibration head\. We expect the construction to be most useful in exactly the settings where labeled in\-distribution data and accelerator hardware are scarce, on\-device monitoring, brain\-computer\-interface calibration, and biomedical informatics among them, and we release a single Python package implementing the full pipeline\.

## Reproducibility

All experiments are deterministic given the input signals and a fixed random seed \(used only for bootstrap resampling\)\. Datasets are public: Sleep\-EDF \(downloaded via themnePython package\), BCI\-IV\-2a \(viamoabb\), MIT\-BIH arrhythmia and the MNE\-sample ERP data \(via PhysioNet andmne\), and ESC\-50\. Compute: a single 2024 Apple\-silicon CPU runs every experiment in this paper in minutes, with no GPU\.

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