Local Reference Geometry Residual Augmentation for Imbalanced Time Series Classification

# Local Reference Geometry Residual Augmentation for Imbalanced Time Series Classification
Source: [https://arxiv.org/html/2609.00093](https://arxiv.org/html/2609.00093)
Chuanhang Qiu, Yanran Xu, Yue Wang, Anthony BagnallAffiliation:School of Electronics and Computer Science University of Southampton Southampton, United Kingdom Emails: C\.Qiu@soton\.ac\.uk, Y\.Xu@soton\.ac\.uk, Yue\.Wang@soton\.ac\.uk, A\.J\.Bagnall@soton\.ac\.uk

###### Abstract

Imbalanced time series classification is often addressed by changing the training distribution, objective, logits, or final threshold\. These interventions address important biases, yet leave a representation\-level question unmeasured: after minority support is reduced, does a learned feature space remain locally reliable around minority regions? We identify a training\-local geometry failure: under imbalance, minority cases can lie in sparse, rest\-dominated, or mixed feature\-space neighborhoods, even when the representation retains useful global class structure\. To diagnose and repair this failure, we propose Local Reference Geometry \(LRG\), a lightweight post\-hoc feature augmentation module applied between a fixed feature extractor and the classifier head\. Using training features only, LRG measures local exposure and class\-mixture risk, then augments each fixed feature with a standardized signed displacement from nearby training geometry and an LDA\-projected residual summary\. On controlled UCR/Bake Off Redux imbalance benchmarks, paired raw\-versus\-LRG comparisons show gains for learned, pretrained, and fixed representations, including when LRG is combined with training\-level interventions and post\-encoder classifier corrections\. Ablations show that the gain comes from the signed local residual appended to the original feature, rather than from generic prototype distances, affinity features, scalar statistics, or VLAD\-style codes\. Further analyses support the proposed local\-geometry failure hypothesis: minority neighborhoods become increasingly rest\-exposed under imbalance, training\-local risk identifies error\-prone regions, and LRG gains concentrate in those high\-risk regions\.

###### Index Terms:

imbalanced time series classification, representation learning

## IIntroduction

Many labeled time series datasets include rare classes that correspond to practically important events, such as equipment faults, medical abnormalities, or other safety\-critical anomalies\. The resulting class imbalance limits minority support during training and can make rare\-event recognition unreliable\. Existing approaches address this problem at different stages of the pipeline\. Data\-level methods alter the training distribution, but require access to the input series and rely on assumptions about temporal similarity\[[1](https://arxiv.org/html/2609.00093#bib.bib22)\]\. Training\-level methods change the sampler, objective, or training procedure, and usually require retraining the encoder\[[2](https://arxiv.org/html/2609.00093#bib.bib23)\]\. Post\-encoder methods adjust the final decision rule while leaving the feature space fixed\[[3](https://arxiv.org/html/2609.00093#bib.bib9)\]\. These interventions address important parts of the imbalance problem, but they do not test whether the learned representation remains locally reliable for recognizing minority samples when training support is scarce\.

Long\-tailed recognition\[[4](https://arxiv.org/html/2609.00093#bib.bib8)\]separates classifier\-level correction from representation\-level effects in computer vision: decoupling studies show that classifier correction can be effective after a representation has been learned\[[5](https://arxiv.org/html/2609.00093#bib.bib16)\], while representation\-learning studies show that imbalance can also distort tail\-class feature geometry\[[6](https://arxiv.org/html/2609.00093#bib.bib17)\]\. This raises the corresponding question for imbalanced TSC: given an existing time\-series representation, can we identify and correct local minority unreliability without retraining the encoder?

In this post\-hoc setting, the relevant unreliability is local to the learned representation\. A minority sample may lie in a region where nearby training support is sparse, mixed across classes, or dominated by non\-minority samples, even when the feature space still separates classes at a coarser scale\. A reweighted or calibrated classifier can adjust priors or thresholds, but it still operates on the features already available from the representation\. We study this missing local information as a training\-local geometry failure\. It can be measured from the training set alone by comparing minority exposure with the training class distribution and by estimating class mixing in local regions\.

To address this training\-local failure after feature extraction, we propose Local Reference Geometry \(LRG\), a residual augmentation method for fixed representations\. LRG is a supervised post\-hoc feature augmentation method that uses training features and labels while leaving the encoder or transform unchanged\. It builds local reference regions in the training representation, then augments each feature vector with a signed local residual and a compact supervised residual summary\. These added features provide information about local training support that may not be explicit in the original representation\. The same reference regions also provide diagnostics of local exposure and class\-mixture risk\. LRG is therefore useful when features have already been produced by a learned encoder, a pretrained model, or a fixed transform, and retraining the feature extractor is costly or impractical\.

Our contributions are:

- •We define and measure training\-local geometry failure for imbalanced TSC using minority\-neighborhood exposure and training\-local class\-mixture risk\.
- •We propose LRG, a supervised post\-hoc residual augmentation method fitted from training features and labels\. LRG augments each fixed feature vector with a signed local residual and an LDA\-projected residual summary, while leaving the encoder or transform unchanged\.
- •We evaluate LRG on controlled UCR/Bake Off Redux multi\-ratio tasks, showing paired gains across learned, pretrained, and fixed representations, compatibility with imbalance interventions and post\-encoder corrections, and larger gains in regions identified as high risk by training\-only diagnostics\.

Code and reproducibility files are provided in the associated repository111https://anonymous\.4open\.science/r/LRG\.

## IIRelated Work

Related work falls into three groups: data\-level and training\-level remedies for imbalanced TSC, post\-encoder decision rules and long\-tailed correction, and feature\-space geometry or local diagnostic representations\. LRG is applied after feature extraction and is complementary to these intervention points\.

### II\-AImbalanced Time Series Remedies

Imbalanced time series classification methods are often described using a coarser data\-level versus algorithm\-level taxonomy\[[1](https://arxiv.org/html/2609.00093#bib.bib22),[2](https://arxiv.org/html/2609.00093#bib.bib23)\]\. Under the intervention\-point view used here, data\-level remedies change the training set through under\-sampling, over\-sampling, or synthesis, including SMOTE\[[7](https://arxiv.org/html/2609.00093#bib.bib7)\], time\-series variants such as e\-SMOTE\[[8](https://arxiv.org/html/2609.00093#bib.bib31)\]and T\-SMOTE\[[9](https://arxiv.org/html/2609.00093#bib.bib21)\]\. Training\-level remedies change the sampler, learner, objective, or training\-time logits, including focal loss\[[10](https://arxiv.org/html/2609.00093#bib.bib13)\], Balanced Softmax\[[11](https://arxiv.org/html/2609.00093#bib.bib15)\], LDAM\[[12](https://arxiv.org/html/2609.00093#bib.bib14)\], and supervised or balanced contrastive objectives\[[6](https://arxiv.org/html/2609.00093#bib.bib17)\]\. These baselines act before or during encoder/classifier training\. LRG instead asks whether an already produced feature space contains minority\-relevant local failures that can be exposed to a downstream head without retraining the encoder\.

### II\-BLong\-Tailed and Post\-Encoder Correction

Long\-tailed recognition in computer vision separates two effects of class imbalance\. Decoupling and calibration studies show that, once a representation is fixed, correcting the classifier head can be a strong intervention; balanced representation learning studies show that imbalance can also shape tail\-class feature geometry\[[4](https://arxiv.org/html/2609.00093#bib.bib8),[5](https://arxiv.org/html/2609.00093#bib.bib16),[6](https://arxiv.org/html/2609.00093#bib.bib17)\]\. This distinction is useful for LRG: classifier correction is a strong frozen\-feature baseline, and the relevant question is whether an additional local residual coordinate still provides information after the decision rule has been corrected\. Recent time series work has begun to develop related representation\-level objectives\[[13](https://arxiv.org/html/2609.00093#bib.bib24),[14](https://arxiv.org/html/2609.00093#bib.bib25)\]\. LRG is aligned with this representation\-level view, but it is post\-hoc: the feature extractor is held fixed and the correction is expressed as an added feature coordinate\.

Post\-encoder methods also operate after features have been extracted, but they act on the classifier head or decision rule\. Logit adjustment\[[3](https://arxiv.org/html/2609.00093#bib.bib9)\], Prior2Posterior\[[15](https://arxiv.org/html/2609.00093#bib.bib11)\], temperature or vector scaling\[[16](https://arxiv.org/html/2609.00093#bib.bib10)\], classifier re\-training, and classifier weight normalization\[[5](https://arxiv.org/html/2609.00093#bib.bib16)\]correct priors, logits, calibration, or classifier weights\. Under a frozen\-feature constraint, these methods are close baselines and we evaluate them in Table[IV](https://arxiv.org/html/2609.00093#S6.T4)\. Their role is different from LRG: they change how existing coordinates are read, whereas LRG exposes a coordinate describing how a sample is displaced relative to nearby training support\. This is why our post\-encoder study uses paired comparisons within each correction rule, testing whether LRG remains useful when a classifier\-level correction is already applied\.

### II\-CFeature\-Space Geometry and Diagnostic Representations

At a computational level, LRG is close to methods that introduce anchors, prototypes, or projections in feature space\. Prototype, metric\-learning, and discriminant methods learn class prototypes, metrics, or projections for classification\[[17](https://arxiv.org/html/2609.00093#bib.bib18),[18](https://arxiv.org/html/2609.00093#bib.bib19),[19](https://arxiv.org/html/2609.00093#bib.bib20)\]\. Radial\-basis features, local coordinate coding, and VLAD\-style residual encodings similarly represent inputs through affinities, local codes, or aggregated residuals from a codebook\[[20](https://arxiv.org/html/2609.00093#bib.bib28),[21](https://arxiv.org/html/2609.00093#bib.bib29),[22](https://arxiv.org/html/2609.00093#bib.bib30)\]\. These methods are designed as general classifiers, projections, or feature encodings\. LRG differs in problem setting: it does not learn a prototype classifier, a metric, or a new representation\. Instead, it asks whether local reference geometry can expose minority\-relevant residual information after a representation has already been learned\. This motivates the local\-feature controls against prototype distances,kk\-means distances, RBF affinities, scalar local statistics, and a VLAD\-style encoding\.

Local\-neighborhood diagnostics characterize class overlap, instance hardness, and minority difficulty from neighborhood composition, including data\-complexity measures,kk\-disagreeing\-neighbor statistics, and ADASYN\[[23](https://arxiv.org/html/2609.00093#bib.bib32),[24](https://arxiv.org/html/2609.00093#bib.bib34),[25](https://arxiv.org/html/2609.00093#bib.bib33)\]\. LRG shares this training\-neighborhood view but uses it for a different purpose: relative rest exposure normalizes local composition by the training prior, and the signed residual uses the same local frame to produce a repair coordinate\. Thus, LRG is best viewed as post\-hoc feature\-space repair: it is computationally related to anchor\-based feature encodings, but it targets the training\-local reliability of a fixed representation and complements sampling, loss\-level, and classifier\-level corrections\.

## IIIProblem Setup

### III\-ATask Definition and Evaluation Metrics

We consider an imbalanced time series task with labeled training split𝒟tr=\{\(xi,yi\)\}i=1n\\mathcal\{D\}\_\{\\mathrm\{tr\}\}=\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{n\}and evaluation split𝒟te\\mathcal\{D\}\_\{\\mathrm\{te\}\}\. A representation extractor maps each time series to a feature vectorz=f⁡\(x\)∈ℝdz=f\(x\)\\in\\mathbb\{R\}^\{d\}\. The extractor may be a fixed transform or a learned encoder\. The classifier head is trained on training features and evaluated on features from𝒟te\\mathcal\{D\}\_\{\\mathrm\{te\}\}\. For each dataset\-ratio task, the minority class is defined from the training split:

ymin=arg⁡minc​\|\{i:yi=c\}\|\.y\_\{\\min\}=\\arg\\min\_\{c\}\|\\\{i:y\_\{i\}=c\\\}\|\.\(1\)Our benchmark keeps the original label space\. In a multi\-class source dataset, onlyyminy\_\{\\min\}is treated as the minority class; the remaining classes are kept as separate classes during head training rather than being collapsed into a single negative class\. Minority\-oriented metrics are then computed by binarizing predictions intoyminy\_\{\\min\}versus rest\. Sens\. denotes minority recall, Spec\. denotes specificity, and Min\-F1 is the F1 score ofyminy\_\{\\min\}\.

### III\-BTraining\-Local Geometry Diagnostics

To quantify training\-local reliability, we measure how often minority samples are surrounded by non\-minority examples in the learned feature space\. For each minority training sampleii, letqiq\_\{i\}be the fraction of itsk=5k=5nearest other training samples that do not belong toyminy\_\{\\min\}\. The raw local rest\-neighbor fraction is𝔼i:yi=ymin\[qi\]\\mathbb\{E\}\_\{i:y\_\{i\}=y\_\{\\min\}\}\[q\_\{i\}\]\. This raw fraction depends on the rest\-class frequency of the specific imbalanced split\. We therefore normalize it byprest=nrest/\(ntr−1\)p\_\{\\mathrm\{rest\}\}=n\_\{\\mathrm\{rest\}\}/\(n\_\{\\mathrm\{tr\}\}\-1\), the random\-neighbor expectation for a minority query under the same training split, wherenrestn\_\{\\mathrm\{rest\}\}is the number of non\-minority training samples, and define relative rest exposure \(RRE\) as

RRE=𝔼i:yi=ymin\[qi\]prest\.\\mathrm\{RRE\}=\\frac\{\\mathbb\{E\}\_\{i:y\_\{i\}=y\_\{\\min\}\}\[q\_\{i\}\]\}\{p\_\{\\mathrm\{rest\}\}\}\.\(2\)RRE equals one when the local rest\-neighbor rate matches the split\-specific rest\-class prior; larger values indicate that minority neighborhoods are more rest\-exposed than expected from the prior alone\. For a collection of dataset\-ratio tasks, RRE is computed within each task first and then averaged across tasks; it is not the ratio of an averaged local fraction to an averaged global prior\. All neighborhood diagnostics are computed only from training features and labels\.

## IVLocal Reference Geometry

LRG uses one training\-local reference frame for both diagnosis and repair\. Scalar outputs, including exposure and class\-mixture risk, diagnose locally unreliable regions; vector outputs provide signed residual coordinates for repair\. This section defines the repair feature, while the analysis section uses the same frame to test whether risky regions coincide with raw prediction failures\.

### IV\-ALocal Reference Regions

Letffdenote a fixed feature extractor; learned encoders are trained before LRG is fit and are not updated by LRG\. LRG operates on flattened features that are standardized with training\-split statistics and thenℓ2\\ell\_\{2\}\-normalized\. LetZtr=\{zi=f⁡\(xi\):\(xi,yi\)∈𝒟tr\}Z\_\{\\mathrm\{tr\}\}=\\\{z\_\{i\}=f\(x\_\{i\}\):\(x\_\{i\},y\_\{i\}\)\\in\\mathcal\{D\}\_\{\\mathrm\{tr\}\}\\\}be the resulting training features\. LRG partitionsZtrZ\_\{\\mathrm\{tr\}\}intoKeffK\_\{\\mathrm\{eff\}\}kk\-means regions\[[26](https://arxiv.org/html/2609.00093#bib.bib26)\]:

\{c1,…,cKeff\}=k​\-means​\(Ztr,Keff\)\.\\\{c\_\{1\},\\ldots,c\_\{K\_\{\\mathrm\{eff\}\}\}\\\}=k\{\\text\{\-means\}\}\(Z\_\{\\mathrm\{tr\}\},K\_\{\\mathrm\{eff\}\}\)\.\(3\)We useKeff=min⁡\(K,max⁡\(2,round⁡\(ntr\)\),ntr\)K\_\{\\mathrm\{eff\}\}=\\min\(K,\\max\(2,\\mathrm\{round\}\(\\sqrt\{n\_\{\\mathrm\{tr\}\}\}\)\),n\_\{\\mathrm\{tr\}\}\)with default capK=8K=8\. Each region stores its center, a radiusρk\\rho\_\{k\}given by the median assigned\-sample distance with floorϵ=10−3\\epsilon=10^\{\-3\}, and its assigned training members\. Labels are used only for the LDA residual and diagnostic class\-composition analyses, not for thekk\-means regions\.

kk\-means serves as a stable, training\-only anchoring mechanism rather than a class model\. Its centers are not minority prototypes, and LRG does not require minority\-dominated anchors\. A rest\-dominated region can itself expose the failure mode: the signed residual describes how a sample departs from the nearby training frame, while the downstream head learns whether that displacement is discriminative\. We avoid local covariance models or label\-aware regions because minority support is scarce under imbalance and high\-dimensional local covariance estimates can be unstable\. The default frame therefore stores only centers and scalar radii, avoids test\-distribution information, and reserves labels for the LDA projection and diagnostic risk analysis\.

### IV\-BLocal Affinity and Residual

For any featurezz, define its soft affinity to local reference regions with temperatureτ\>0\\tau\>0:

ak​\(z\)=exp\(−∥z−ck∥22/τ\)∑jexp\(−∥z−cj∥22/τ\)\.a\_\{k\}\(z\)=\\frac\{\\exp\(\-\\\|z\-c\_\{k\}\\\|\_\{2\}^\{2\}/\\tau\)\}\{\\sum\_\{j\}\\exp\(\-\\\|z\-c\_\{j\}\\\|\_\{2\}^\{2\}/\\tau\)\}\.\(4\)The local reference center and scale are

μ⁡\(z\)=∑k=1Keffak​\(z\)​ck,σ⁡\(z\)=∑k=1Keffak​\(z\)​ρk\.\\mu\(z\)=\\sum\_\{k=1\}^\{K\_\{\\mathrm\{eff\}\}\}a\_\{k\}\(z\)c\_\{k\},\\quad\\sigma\(z\)=\\sum\_\{k=1\}^\{K\_\{\\mathrm\{eff\}\}\}a\_\{k\}\(z\)\\rho\_\{k\}\.\(5\)The local residual is

rfull​\(z\)=z−μ⁡\(z\)max⁡\(σ⁡\(z\),ϵ\)\.r\_\{\\mathrm\{full\}\}\(z\)=\\frac\{z\-\\mu\(z\)\}\{\\max\(\\sigma\(z\),\\epsilon\)\}\.\(6\)This residual is a signed local\-reference coordinate, not a prototype decision rule\. Soft affinities smooth hard cluster boundaries and let interface samples inherit a reference center from several nearby anchors\. The radial scale makes residual magnitudes comparable across dense and diffuse regions, while the sign and direction describe how the feature departs from the local training frame\.

### IV\-CGeometric Effect of Residual Augmentation

The role of the residual can be seen by comparing the decision functions available to a linear downstream head\. A linear score on the raw feature has the form

syraw​\(z\)=wy⊤​z\+by,s\_\{y\}^\{\\mathrm\{raw\}\}\(z\)=w\_\{y\}^\{\\top\}z\+b\_\{y\},\(7\)so pairwise class decisions have global affine boundaries in the original feature space\. With residual augmentation, the score becomes

syLRG​\(z\)=wy⊤​z\+vy⊤​rfull​\(z\)\+uy⊤​rlda​\(z\)\+by\.s\_\{y\}^\{\\mathrm\{LRG\}\}\(z\)=w\_\{y\}^\{\\top\}z\+v\_\{y\}^\{\\top\}r\_\{\\mathrm\{full\}\}\(z\)\+u\_\{y\}^\{\\top\}r\_\{\\mathrm\{lda\}\}\(z\)\+b\_\{y\}\.\(8\)Settingvy=uy=0v\_\{y\}=u\_\{y\}=0recoverssyraw​\(z\)s\_\{y\}^\{\\mathrm\{raw\}\}\(z\), so residual augmentation does not remove any decision function available onzrawz\_\{\\mathrm\{raw\}\}\. The additional terms are not arbitrary nonlinear capacity; they are tied to local displacement from the training reference frame\. For intuition, consider the hard\-assignment limit in which the nearest regionkkhasak​\(z\)→1a\_\{k\}\(z\)\\to 1and all other regions haveaj​\(z\)→0a\_\{j\}\(z\)\\to 0forj≠kj\\neq k\. Then the soft local mean and scale reduce toμ⁡\(z\)≈ck\\mu\(z\)\\approx c\_\{k\}andσ⁡\(z\)≈ρk\\sigma\(z\)\\approx\\rho\_\{k\}, sorfull​\(z\)≈\(z−ck\)/ρkr\_\{\\mathrm\{full\}\}\(z\)\\approx\(z\-c\_\{k\}\)/\\rho\_\{k\}\. Ignoring the low\-dimensional LDA term for the moment,

syLRG​\(z\)≈\(wy\+vyρk\)⊤​z\+by−vy⊤​ckρk\.s\_\{y\}^\{\\mathrm\{LRG\}\}\(z\)\\approx\\left\(w\_\{y\}\+\\frac\{v\_\{y\}\}\{\\rho\_\{k\}\}\\right\)^\{\\top\}z\+b\_\{y\}\-\\frac\{v\_\{y\}^\{\\top\}c\_\{k\}\}\{\\rho\_\{k\}\}\.\(9\)Thus, within a local reference region, the same linear head can express a region\-conditioned affine correction; with soft affinities, this correction is smooth rather than a hard set of independent classifiers\. The LDA term has the same affine form because it is an affine function ofrfull​\(z\)r\_\{\\mathrm\{full\}\}\(z\), as shown below\. This distinguishes LRG from scalar local statistics: affinity, purity, or reliability can mark a region as near, mixed, or sparse, but a signed residual specifies the direction of deviation from the local training frame\. The downstream head remains lightweight: LRG shares one downstream head, rather than introducing a neural head or region\-specific classifiers\. Under the paired protocol, gains after adding LRG reflect information exposed by the residual frame rather than a wholesale change in model class\.

### IV\-DLDA\-Projected Residual

The residual can be high\-dimensional, and not every direction is class\-relevant\. We therefore summarize residuals with shrinkage\-regularized LDA directionsWLDAW\_\{\\mathrm\{LDA\}\}fit on training residuals and labels\. The projection is applied to residuals rather than raw features: raw\-feature LDA would mainly recover global class directions that a linear head can already learn, whereas residual\-space LDA identifies class\-aligned directions of local deviation\. This LDA fit is global over the training residuals, not separate within eachkk\-means reference region, so it does not require estimating class\-conditional covariance matrices inside sparse local clusters\. With training\-split meanmLDAm\_\{\\mathrm\{LDA\}\}and diagonal standard deviation matrixDLDAD\_\{\\mathrm\{LDA\}\}, the projected residual is

pLDA​\(z\)\\displaystyle p\_\{\\mathrm\{LDA\}\}\(z\)=WLDA⊤​rfull​\(z\),\\displaystyle=W\_\{\\mathrm\{LDA\}\}^\{\\top\}r\_\{\\mathrm\{full\}\}\(z\),\(10\)rlda​\(z\)\\displaystyle r\_\{\\mathrm\{lda\}\}\(z\)=DLDA−1​\(pLDA​\(z\)−mLDA\)\.\\displaystyle=D\_\{\\mathrm\{LDA\}\}^\{\-1\}\\left\(p\_\{\\mathrm\{LDA\}\}\(z\)\-m\_\{\\mathrm\{LDA\}\}\\right\)\.Sincerlda​\(z\)r\_\{\\mathrm\{lda\}\}\(z\)is affine inrfull​\(z\)r\_\{\\mathrm\{full\}\}\(z\), adding it preserves the local affine interpretation above\. For binary tasks this is one\-dimensional; for aCC\-class task it can contain at mostC−1C\-1directions\. We use a shrinkage LDA solver \(“lsqr” with automatic shrinkage\)\. All standardization, LDA fitting, projection normalization, and head fitting are training\-only\. In all main experiments, we use fixed local geometry parameters:K=8K=8, retained LDA dimensionq=1q=1, andτ=1\.0\\tau=1\.0\. Becauserlda​\(z\)r\_\{\\mathrm\{lda\}\}\(z\)is appended to\[zraw,rfull​\(z\)\]\[z\_\{\\mathrm\{raw\}\},r\_\{\\mathrm\{full\}\}\(z\)\], the one\-dimensional default acts as a low\-variance supervised residual hint rather than a bottleneck or independent nonlinear map\.

### IV\-ELRG Residual Augmentation and Complexity

The final feature passed to the downstream head is

zLRG=\[zraw,rfull​\(z\),rlda​\(z\)\]\.z\_\{\\mathrm\{LRG\}\}=\[z\_\{\\mathrm\{raw\}\},r\_\{\\mathrm\{full\}\}\(z\),r\_\{\\mathrm\{lda\}\}\(z\)\]\.\(11\)LRG then trains the same lightweight heads used forzrawz\_\{\\mathrm\{raw\}\}, mainly class\-weighted logistic regression and threshold\-tuned logistic regression\. The LRG reference frame–the standardizer, centers, radii, and LDA projection–is built using only training features and labels, and is then kept fixed\. At evaluation time, each feature is standardized with training statistics, assigned soft affinities to the stored centers, converted into residual coordinates, and passed to the fitted head\. The procedure uses no test labels and estimates no test\-set prior, matching the information constraint of the post\-encoder baselines\.

Letnnbe the number of training samples,ddthe feature dimension,CCthe number of classes,IIthekk\-means iteration count, andq≤C−1q\\leq C\-1\. Reference construction costsO⁡\(I​n​Keff​d\)O\(InK\_\{\\mathrm\{eff\}\}d\), and the persistent LRG state storesO⁡\(Keff​d\+d​q\)O\(K\_\{\\mathrm\{eff\}\}d\+dq\)values plus the feature standardizer and linear head\. At test time, each sample costsO⁡\(Keff​d\+d​q\)O\(K\_\{\\mathrm\{eff\}\}d\+dq\)for the local frame and LDA residual, plusO⁡\(C⁡\(2​d\+q\)\)O\(C\(2d\+q\)\)for the linear head\. Once features are extracted, this overhead is independent of the original series length: LRG stores no raw series, pairwise training distances, or nearest\-neighbor graphs at inference time\.

## VExperiments

The experiments evaluate LRG through paired raw\-versus\-LRG comparisons and compatibility checks with imbalance interventions and post\-encoder corrections\. In each paired comparison, the representation, train/test split, and classifier protocol are held fixed, and only the LRG residual augmentation is added\. This separates the main question, whether local residuals improve an available representation, from absolute classifier ranking\.

### V\-ABenchmark Construction

We construct a controlled univariate imbalanced benchmark from the fixed train/test splits of the UCR archive\[[27](https://arxiv.org/html/2609.00093#bib.bib12)\]and 30 univariate classification datasets released with Bake Off Redux\[[28](https://arxiv.org/html/2609.00093#bib.bib3)\]\. Following prior imbalanced TSC benchmark construction\[[9](https://arxiv.org/html/2609.00093#bib.bib21)\], we keep the original binary or multiclass label space and downsample only one training class\. Letnmaxn\_\{\\max\}be the largest class count in the original training split and letyminy\_\{\\min\}denote the selected minority class\. For target ratioR:1R\{:\}1, we keep⌊nmax/R⌋\\lfloor n\_\{\\max\}/R\\rfloortraining samples fromyminy\_\{\\min\}and retain all other training samples\. The test split is left unchanged to isolate minority training\-support scarcity while preserving the fixed evaluation problem\.

The paired comparisons use the full3:13\{:\}1/5:15\{:\}1/10:110\{:\}1/20:120\{:\}1scope, giving 138 dataset\-ratio tasks per backbone\. A dataset\-ratio pair is included only when the original training split can realize the requested ratio, so fewer datasets are available at higher imbalance\. We provide the retained and excluded dataset\-ratio lists on the associated repository\.

TABLE I:Summary statistics for the materialized imbalanced benchmark\. Each column reports the retained dataset\-ratio tasks for the target training ratio\. Except for dataset counts, entries are median \(min\-max\)\.
### V\-BRepresentations and Heads

We evaluate LRG across learned encoders, frozen pretrained features, and randomised feature\-transform representations\. Learned encoders include InceptionTime\-style networks\[[29](https://arxiv.org/html/2609.00093#bib.bib4)\], ResNet1D, TCN, TimesNet\[[30](https://arxiv.org/html/2609.00093#bib.bib5)\], a classification\-oriented PatchTST\-style encoder\[[31](https://arxiv.org/html/2609.00093#bib.bib6)\], and MLP encoders\. All learned encoders use the same inner validation split, early stopping criterion, and plateau scheduling; weighted variants additionally use class\-weighted cross entropy and balanced sampling where specified\. We also evaluate frozen MOMENT features\[[32](https://arxiv.org/html/2609.00093#bib.bib27)\]and features from the combined MultiRocket\-Hydra randomised transform\[[33](https://arxiv.org/html/2609.00093#bib.bib1),[34](https://arxiv.org/html/2609.00093#bib.bib2)\]\.

Classifier heads are chosen to match the representation setting\. The learned\-encoder, pretrained\-feature, and intervention studies use class\-weighted logistic regression\. The MultiRocket\-Hydra feature\-transfer row uses threshold\-tuned logistic regression\. The post\-encoder correction study fixes the Inception feature representation and compares calibrated, threshold\-tuned, or retrained classifier heads with and without LRG augmentation\.

### V\-CPaired Protocol and Baseline Groups

For a given representation, train/test split, and classifier head, the paired protocol compareszrawz\_\{\\mathrm\{raw\}\}with\[zraw,rfull​\(z\),rlda​\(z\)\]\[z\_\{\\mathrm\{raw\}\},r\_\{\\mathrm\{full\}\}\(z\),r\_\{\\mathrm\{lda\}\}\(z\)\]\. The task, data split, representation, and classifier protocol are therefore shared; only the LRG residual augmentation is added\. This makes each comparison a direct test of whether the local residual features improve the available representation\.

We use three paired studies\. Representation Evaluation \(Table[II](https://arxiv.org/html/2609.00093#S6.T2)\) applies the protocol across learned encoders, pretrained features, and randomised feature\-transform representations\. Imbalance\-Intervention Compatibility \(Table[III](https://arxiv.org/html/2609.00093#S6.T3)\) applies the same Base\-versus\-\+LRG comparison after losses, samplers, or SMOTE\-style over\-sampling have changed the encoder training or training data\. Post\-Encoder Correction Compatibility \(Table[IV](https://arxiv.org/html/2609.00093#S6.T4)\) fixes the Inception feature representation and tests whether residual augmentation remains useful under classifier\-level corrections\.

The residual\-signal and parameter studies isolate which components carry the gain, while the analysis section tests whether the gains align with measurable training\-local geometry failure\. Balanced accuracy and Macro\-F1 are the primary cross\-task metrics; Min\-F1, Sens\., and Spec\. are used for minority\-specific questions\.

## VIExperimental Results

### VI\-ARepresentation Evaluation

TABLE II:Representation evaluation\. W/T/L counts per\-task wins, ties, and losses for \+LRG over the corresponding raw feature\. Parentheses in W/T/L cells show Wilcoxon signed\-rank evidence levels \(∗\*:p<0\.05p<0\.05,∗⁣∗\*\*:p<0\.01p<0\.01,∗∗∗\*\*\*:p<0\.001p<0\.001\)\.Table[II](https://arxiv.org/html/2609.00093#S6.T2)reports the paired representation evaluation study\. The experiment is not designed as a classifier comparison; it tests whether LRG improves a given representation under the same split and classifier protocol\. The evidence for LRG is therefore the within\-row change fromzrawz\_\{\\mathrm\{raw\}\}tozLRGz\_\{\\mathrm\{LRG\}\}\.

Table[II](https://arxiv.org/html/2609.00093#S6.T2)uses the representation\-evaluation training configuration, where learned encoders are trained with balanced mini\-batch sampling\. In contrast, the Nat\. rows in Table[III](https://arxiv.org/html/2609.00093#S6.T3)use ordinary CE with the original imbalanced dataloader\. The two Inception baselines therefore evaluate different training protocols and are not expected to have identical raw performance\.

LRG improves every representation setting in this study\. Balanced accuracy increases by 1\.3\-5\.0 percentage points and Macro\-F1 by 2\.3\-7\.7 percentage points\. The largest gains occur for weaker raw feature spaces: TimesNet and TCN gain more than four points in balanced accuracy and more than five points in Macro\-F1, while frozen MOMENT gains five points in balanced accuracy and 7\.7 points in Macro\-F1\. Inception has the strongest raw balanced accuracy and still improves, showing that the effect is not restricted to weak representations\. The frozen MOMENT row captures the post\-hoc setting most directly: the pretrained encoder supplies fixed features, and LRG augments those features before the classifier is fitted, without updating the encoder\. The W/T/L counts show that the gains are broadly distributed across tasks: \+LRG wins more tasks than it loses for every learned and frozen row on both metrics, with all such rows significant atp<0\.001p<0\.001\. The randomized\-transform row shows the same direction: MultiRocket\-Hydra\+LRG improves both average metrics, with stronger evidence for Macro\-F1 and weaker but positive evidence for balanced accuracy\.

### VI\-BLRG with Imbalance\-Aware Methods

We next ask whether LRG remains useful after standard imbalance\-aware methods have already modified the training data or objective\. Table[III](https://arxiv.org/html/2609.00093#S6.T3)pairs each baseline setting with the same setting followed by LRG residual augmentation\. The classifier is class\-weighted logistic regression for this set of experiments, so the Base\-versus\-\+LRG difference reflects feature augmentation under a fixed decision rule rather than threshold calibration\.

TABLE III:LRG with standard imbalance\-aware methods\. For this compact table only, Bal\. denotes balanced accuracy, MacF1 denotes Macro\-F1, and MinF1 denotes Min\-F1\.Rows are paired by baseline setting, so the LRG delta is the within\-pair difference; bold marks the better value in each pair\. The Avg\. block reports the mean across the three backbone columns\. BS denotes Balanced Softmax\[[11](https://arxiv.org/html/2609.00093#bib.bib15)\], LDAM denotes LDAM\-DRW\[[12](https://arxiv.org/html/2609.00093#bib.bib14)\], and Nat\. denotes ordinary CE\. Nat\., BS, and LDAM are trained on the original imbalanced dataloader; SMOTE and e\-SMOTE are input\-space over\-sampling methods paired with weighted CE\.

LRG improves all baseline settings and backbones\. In the Avg\. block, adding LRG improves balanced accuracy by 2\.9\-3\.7 percentage points, Macro\-F1 by 3\.7\-4\.5 points, and Min\-F1 by 3\.6\-4\.7 points\. The gains are largest for TCN, where the raw representation is less stable: across imbalance\-aware methods, TCN gains 4\.3\-5\.1 points in balanced accuracy, 5\.4\-6\.3 points in Macro\-F1, and 5\.5\-6\.9 points in Min\-F1\. The improvements remain visible for ResNet1D and Inception, including after Balanced Softmax, LDAM, SMOTE, and e\-SMOTE\. These results suggest that imbalance\-aware losses and over\-sampling can improve the learned representation while still leaving useful local residual information for LRG to exploit\.

\(a\) Balanced accuracy

\(b\) Macro\-F1

\(c\) Min\-F1

Fig\. 1:Average rank diagrams for the intervention rows in Table[III](https://arxiv.org/html/2609.00093#S6.T3)\.Figure[1](https://arxiv.org/html/2609.00093#S6.F1)ranks the methods over the same 138 backbone–dataset–ratio problems as Table[III](https://arxiv.org/html/2609.00093#S6.T3)\. Lower rank is better\. Cliques indicate methods that are not separated by a one\-sided Wilcoxon signed\-rank test with Holm correction atα=0\.1\\alpha=0\.1\. The leading group consists of \+LRG variants; small rank reversals between BS\+LRG and LDAM\+LRG should be read as metric\-specific differences rather than as a stable ordering between the two losses\.

### VI\-CPost\-Encoder Classifier Corrections

We next fix the Inception feature representation and ask whether LRG remains useful when the classifier has already been corrected\. Thresholds, calibration parameters, and predicted\-prior estimates are selected using training\-only out\-of\-fold \(OOF\) predictions, with no access to test labels or the test distribution\. For \+LRG, all LRG components are fit within each OOF training fold\. Unlike Table[III](https://arxiv.org/html/2609.00093#S6.T3), which changes the training data or objective, Table[IV](https://arxiv.org/html/2609.00093#S6.T4)changes only the downstream decision rule\. The key comparison is the paired Base\-versus\-\+LRG difference within each correction\.

TABLE IV:Compatibility with post\-encoder classifier corrections on Inception features\.Base denotes each correction fitted on raw Inception features, while \+LRG fits the same correction type on LRG\-augmented features under the same training\-only protocol\. The Level column identifies which part of the frozen\-feature decision pipeline is changed\. Threshold tuning selects a minority threshold from out\-of\-fold probabilities\. LogitAdj applies training\-prior logit adjustment\[[3](https://arxiv.org/html/2609.00093#bib.bib9)\]; P2P replaces the empirical prior with the out\-of\-fold predicted prior of the model, following the Prior2Posterior correction idea\[[15](https://arxiv.org/html/2609.00093#bib.bib11)\]\. VectorScale is a regularized class\-specific affine scaling of logits inspired by post\-hoc calibration methods\[[16](https://arxiv.org/html/2609.00093#bib.bib10)\]\. TauNorm rescales classifier weights, and cRT retrains only the classifier with class\-balanced weighting, following decoupled classifier correction ideas\[[5](https://arxiv.org/html/2609.00093#bib.bib16)\]\.

LRG improves balanced accuracy, Macro\-F1, and Min\-F1 for every post\-encoder correction in this table\. The largest deltas occur for TauNorm, whose raw row is the weakest in the table, but the more informative result is that LRG also improves stronger classifier\-level corrections: VectorScale gains 2\.16 points in balanced accuracy, 2\.77 points in Macro\-F1, and 2\.15 points in Min\-F1, while cRT gains 1\.82, 2\.59, and 1\.85 points, respectively\. Thus, classifier\-level correction and residual augmentation are complementary: the former changes the decision rule on fixed features, whereas LRG exposes additional local residual coordinates before the head is fit\.

### VI\-DResidual Signal and Sensitivity

This subsection asks whether LRG gains arise specifically from signed local residuals rather than from generic local descriptors, and then examines the contribution and parameter sensitivity of the residual coordinates\. All tables use the same 138 dataset–ratio tasks and the same cached Inception CE representations\. Reported values are paired percentage\-point changes relative to raw features under the same downstream head\. For each configuration, we first average the class\-weighted logistic and threshold\-tuned deltas within each task and then average across tasks\. Table[V](https://arxiv.org/html/2609.00093#S6.T5)compares LRG with alternative local features, while Table[VI](https://arxiv.org/html/2609.00093#S6.T6)reports residual ablations and one\-factor\-at\-a\-time parameter sensitivity\.

TABLE V:Local\-feature controls\.TABLE VI:Residual ablation and parameter sensitivity\.Table[V](https://arxiv.org/html/2609.00093#S6.T5)treats the complete LRG residual augmentation as the proposed signal and compares it against local\-feature alternatives\. Prototype distances,kk\-means distances, RBF\-style affinities, and scalar local statistics give small gains, while the VLAD\-style replacement code improves specificity at the cost of sensitivity and the main metrics\. The LRG row improves all main metrics and sensitivity while preserving the original feature\. This supports the directionality principle in Section[IV\-C](https://arxiv.org/html/2609.00093#S4.SS3): local distances and reliability statistics can mark a region as nearby, mixed, or sparse, but they do not expose the signed direction of the sample’s deviation from the local training frame\.

The residual ablation block in Table[VI](https://arxiv.org/html/2609.00093#S6.T6)compares the two residual components and their combination\. Bothrfullr\_\{\\mathrm\{full\}\}andrldar\_\{\\mathrm\{lda\}\}are useful on their own, and appending them together gives the strongest balanced accuracy, Macro\-F1, and Min\-F1\.

The sensitivity blocks are one\-factor\-at\-a\-time robustness checks rather than hyperparameter tuning procedures\. The main experiments use the fixed defaultsK=8K=8,dLDA=1d\_\{\\mathrm\{LDA\}\}=1, andτ=1\.0\\tau=1\.0\. All tested settings retain positive aggregate gains\. Performance varies only modestly acrossK∈\{4,8,16\}K\\in\\\{4,8,16\\\}, whileτ=1\.0\\tau=1\.0gives the strongest result among the tested temperatures\. IncreasingdLDAd\_\{\\mathrm\{LDA\}\}from one to three produces small additional gains on the multiclass tasks\. For binary tasks, all three settings reduce to the single available discriminant direction becausedLDA≤C−1d\_\{\\mathrm\{LDA\}\}\\leq C\-1\. We retaindLDA=1d\_\{\\mathrm\{LDA\}\}=1as a compact, pre\-specified default that avoids dataset\-specific dimensionality selection; we do not claim that it is the empirically optimal value\.

Finally, a descriptive margin decomposition shows that the fitted full\-LRG heads actively use the residual coordinates\. Under standardized features, thezraw/rfull/rldaz\_\{\\mathrm\{raw\}\}/r\_\{\\mathrm\{full\}\}/r\_\{\\mathrm\{lda\}\}blocks account for3\.9%/60\.9%/35\.2%3\.9\\%/60\.9\\%/35\.2\\%of the absolute minority\-versus\-strongest\-rest margin for class\-weighted logistic regression, and3\.6%/61\.7%/34\.8%3\.6\\%/61\.7\\%/34\.8\\%for threshold tuning\. Because these blocks are correlated andrldar\_\{\\mathrm\{lda\}\}is derived fromrfullr\_\{\\mathrm\{full\}\}, the percentages are not unique causal attributions\. The following analysis examines where this residual use improves predictions\.

## VIIAnalysis

The paired experiments above establish the performance effect of LRG\. We next ask whether this effect is tied to the proposed training\-local geometry failure\. The analysis starts with the feature space itself: does imbalance make minority neighborhoods more rest\-exposed? It then tests whether training\-only class\-composition risk identifies error\-prone regions and whether LRG gains are larger in those regions\. We close by examining how downstream heads trade sensitivity against specificity when imbalance becomes severe\.

### VII\-ARQ1: Does Imbalance Alter Local Geometry?

Before applying any correction, we test whether stronger imbalance changes the local geometry around the training\-defined minority class\. We compute RRE according to Eq\.[2](https://arxiv.org/html/2609.00093#S3.E2)before fitting any downstream head\. The neighborhoods are computed after the same training\-split standardization andℓ2\\ell\_\{2\}normalization used by LRG, so they should be read as an operational local\-support diagnostic rather than as a claim that a fixed\-transform classifier itself is nearest\-neighbor based\.

Table[VII](https://arxiv.org/html/2609.00093#S7.T7)compares RRE across imbalance ratios and feature families\. The fixed row summarizes Hydra and MultiRocket fixed\-transform features, the learned row summarizes Inception and PatchTST features trained with weighted CE, and the pretrained row uses frozen MOMENT features\. RRE increases in all three feature families as imbalance becomes more severe\. Because RRE divides the local rest\-neighbor fraction by the split\-specific rest\-class prior, this trend is not the mechanical effect of having more rest\-class samples in the training split\. The fixed and pretrained rows are especially useful controls: their extractors are unchanged as the imbalance ratio varies, so the increase reflects support thinning in the current imbalanced training set\. As minority training samples are removed, minority neighborhoods lose same\-class support and move toward the split\-specific random\-prior baseline, even though the absolute RRE values remain below one\. The learned row has lower absolute RRE, as expected for task\-supervised features trained with labels to be more class\-aligned under this local metric; nevertheless, it shows the same support\-local effect in a feature space also shaped by imbalance\-aware supervised training\. This motivates the next question: whether these entangled local regions correspond to actual prediction errors, and whether LRG residuals are useful precisely where the local training geometry is unreliable\.

TABLE VII:Relative rest exposure \(RRE\) across feature families\.
### VII\-BRQ2: Do Local Risks Track Errors and LRG Gains?

RQ1 shows that stronger imbalance makes minority neighborhoods more rest\-exposed\. We now ask whether the same training\-local frame can identify which regions are error\-prone and whether LRG helps most in those regions\. These risk scores use only class\-composition statistics from training features, before test labels are observed\. Within each local reference region, we score class mixture using the local minority fraction, rest fraction, minority\-versus\-rest ambiguity, and multiclass impurity\. The false\-positive risk is high for rest samples near ambiguous minority support; the false\-negative risk is high for minority samples near ambiguous rest support; and the hybrid risk combines these two directions with region impurity\. A test feature inherits these region scores through the same training\-derived affinities used by the residual\. These scalar diagnostics share LRG’s reference frame but use class composition rather than signed displacement, so enrichment is not guaranteed by construction\. Test labels are used only afterward to check whether high\-risk regions indeed contain more errors\. Using the same feature\-family grouping as Table[VII](https://arxiv.org/html/2609.00093#S7.T7), Table[VIII](https://arxiv.org/html/2609.00093#S7.T8)reports three summaries\. FP enrichment is the high\-to\-low false\-positive ratio after binning by false\-positive risk\. The low/high error columns report total error rates after binning by hybrid risk, and positive FP enrichment excludes tasks whose low\-risk bin has zero false positives\.

TABLE VIII:High\-risk local regions enrich errors under threshold tuning\. Error rates are percentages\.Table[VIII](https://arxiv.org/html/2609.00093#S7.T8)shows that high\-risk regions are not arbitrary partitions\. High FP\-risk regions enrich false positives, especially when conditioning on tasks where false positives exist in the low\-risk reference bin\. High hybrid\-risk regions also have higher total error rates in every feature family\. This supports the diagnostic claim: training\-local geometry can stratify test regions by error risk before test labels are used for evaluation\.

We then test whether the intervention is aligned with this diagnostic\. For each test sample, we compute its training\-only hybrid risk score, split samples into low\- and high\-risk bins within each dataset\-ratio task, and compare raw and LRG predictions within each bin\. Table[IX](https://arxiv.org/html/2609.00093#S7.T9)shows that the high\-risk bin has both higher raw error and a larger paired LRG error reduction\. The gain is therefore not only an average effect over all samples; it is amplified where the local training geometry is more entangled\. The bin\-level decomposition shows that this concentration is especially strong for false\-positive repair, motivating the head\-level safety analysis in RQ3\.

TABLE IX:Risk\-bin gain analysis with class\-weighted logistic\.As a qualitative check, Figure[2](https://arxiv.org/html/2609.00093#S7.F2)visualizes the same Inception CE feature map on Strawberry under 5:1 and 10:1 imbalance\. The upper row shows the minority/rest structure in the raw t\-SNE coordinates\. The lower row overlays LRG error transitions on the same coordinates: green markers are false negatives or false positives corrected by LRG, while purple and black markers are false negatives and false positives that remain after LRG\. The figure does not require global separation in the plotted embedding\. Instead, it shows whether errors in locally mixed regions are corrected on the same map\. At 5:1, LRG makes a smaller but consistent change, reducing false negatives from 25 to 21 and false positives from 14 to 12\. At 10:1, where RRE is higher \(0\.400\.40versus0\.320\.32\), the green false\-negative repairs are more prominent: false negatives decrease from 132 to 40, with false positives increasing from 0 to 3\. This visualizes the risk\-bin result that LRG gains concentrate in locally mixed minority regions\.

![Refer to caption](https://arxiv.org/html/2609.00093v1/tsne.png)Fig\. 2:LRG error repair on Strawberry at 5:1 and 10:1\. The upper row shows minority/rest features; the lower row overlays error transitions on the same t\-SNE coordinates\.
### VII\-CRQ3: High\-Imbalance Behavior and Head\-Level Safety

To isolate ratio effects from changes in which source datasets are averaged, this subsection uses the matched subset of1414datasets that appear under all four ratios, 3:1, 5:1, 10:1, and 20:1\. We summarize LRG behavior by averaging paired deltas over five post\-encoder settings: linear, threshold tuning, logit adjustment, vector scaling, and cRT\. In Table[X](https://arxiv.org/html/2609.00093#S7.T10), RRE is computed on the corresponding Inception CE diagnostic run; the remaining columns are paired mean deltas in percentage points\.

TABLE X:Ratio\-level LRG effect on the matched1414\-dataset scope\.On this matched1414dataset scope, Table[X](https://arxiv.org/html/2609.00093#S7.T10)shows a clear high\-imbalance trend\. RRE rises from0\.2990\.299at 3:1 to0\.4540\.454at 20:1, showing that minority neighborhoods become increasingly rest\-exposed even after normalizing by the global rest\-class prior\. LRG gains increase in the same high\-imbalance regimes\. At 20:1, LRG improves balanced accuracy by\+2\.87\+2\.87pp, Min\-F1 by\+6\.65\+6\.65pp, sensitivity by\+2\.64\+2\.64pp, and specificity by\+2\.91\+2\.91pp on average across the five post\-encoder settings\. Thus, the high\-imbalance advantage is not merely a specificity\-driven false\-positive repair: sensitivity and specificity both improve\. This is consistent with the geometric interpretation\. As imbalance removes minority support, raw local neighborhoods become harder to separate fromzrawz\_\{\\mathrm\{raw\}\}alone; the signed training\-referenced residual then provides a larger corrective signal for the downstream head\. The near\-zero sensitivity deltas at 3:1 and 5:1 are also informative: when the base post\-encoder settings already preserve minority recall reasonably well, LRG improves the local boundary without materially changing the minority\-recall operating point\.

## VIIIDiscussion

Taken together, the results support a focused claim: imbalance induces a measurable training\-local geometry failure in otherwise useful time series representations, and signed local residuals expose corrective information for a lightweight downstream head\. LRG does not treat the raw representation as failed globally\. Instead, it asks whether minority support is locally unreliable after a feature space is already available, then supplies a directional coordinate tied to nearby training geometry\. This post\-hoc setting is practical when pretrained features are expensive to fine\-tune or only features are available\.

This distinction also separates diagnostics from repair\. Reliability, affinity, and risk statistics summarize whether a region is pure, mixed, sparse, or error\-prone, but they do not specify a discriminative direction\. A local residual is a signed coordinate: it tells the head how a sample is displaced relative to nearby training geometry\. In this sense, diagnostics locate the failure, while the residual provides the repair coordinate\.

LRG should therefore be viewed as a representation\-level intervention rather than another sampler, loss, or threshold rule\. Over\-sampling changes the training distribution, class weighting changes the loss, and threshold tuning changes the final decision boundary\. LRG instead exposes a local residual signal that may remain after those mechanisms, while the downstream head remains responsible for the sensitivity–specificity operating point\. This explains why LRG is complementary to long\-tailed losses, over\-sampling, and post\-encoder corrections rather than a replacement for them\. The added post\-encoder cost is modest in our feature\-space experiments: excluding encoder inference, average LRG fitting takes 0\.433s on Inception features and 1\.676s on MOMENT features, with microsecond\-scale prediction overhead\.

Usingkk\-means centers and a scalar radial scale keeps the post\-hoc state small and avoids unstable covariance estimates under sparse minority support, but it also limits the ability to model anisotropic local geometry\. When minority support is extremely sparse or highly diffuse, rest\-dominated anchors may yield unstable residual directions, and the LDA projection can become high\-variance\. The present benchmark is also univariate by design: the fixed UCR and Bake Off Redux splits provide a broad controlled setting for materializing label imbalance, but multivariate archives remain an important next step\. Because LRG operates after feature extraction, the main question there is how channel interactions affect the learned local reference geometry\. These limitations point to adaptive reference construction and training\-time objectives for minority\-reliable local geometry as natural future directions\.

## IXConclusion

We introduced Local Reference Geometry as a diagnostic\-and\-repair framework for imbalanced time series representations\. LRG quantifies training\-local geometry failure through exposure and class\-mixture risk, then appends signed full and LDA\-projected residual coordinates to the existing feature\. Experiments show that this directional residual signal provides useful corrective information across learned, pretrained, and fixed representations, and that the largest gains occur in local regions identified as risky by the same training\-only frame\. Future work can extend this local\-reliability view to training\-time objectives for minority\-reliable representation learning\.

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