TopoExplore: Topological Discrimination for Archive-Based Exploration

# Topological Discrimination for Archive-Based Exploration Preliminary report
Source: [https://arxiv.org/html/2607.09971](https://arxiv.org/html/2607.09971)
\(July 2026 \(v1 — preliminary results; expanded evaluation to follow\)\)

###### Abstract

Archive\-based exploration methods such as Go\-Explore select which visited state to return to using visitation rarity, and frontier methods in robotics return to the boundary of the unknown; neither asks whether the unexplored region behind a boundary is*enterable at all*\. Exploration is not just about finding reward—it is about collecting a structurally complete experience for downstream learning and planning\. We introduce TopoExplore, which augments Go\-Explore cell selection with a periodic topological pass: enclosed unexplored regions \(*voids*\) of the visited\-set occupancy grid are detected by flood fill—exactly theH1H\_\{1\}classes of the binary cubical complex—and a decaying selection bonus is placed only on their*strict entrances*\(gap or door cells\), so sealed regions are never targeted and entered regions retire\. On a controlled 18\-environment MiniGrid suite \(15 seeds, frozen hyperparameters\) TopoExplore attains a 1\.52×\\timesgeometric\-mean speedup in median steps\-to\-first\-entry over its exact Go\-Explore ablation, versus 1\.37×\\timesfor a strong frontier baseline on the same harness; critically, frontier exploration*degrades*precisely when sealed decoy structure appears \(0\.83–1\.48×\\timeson decoy environments vs\. 1\.65–2\.11×\\timesfor TopoExplore\), while TopoExplore holds its largest win on hard multi\-interaction doors \(10\.9×\\times\)\. We report an honest negative on Montezuma’s Revenge—without wall knowledge, unreachable occupancy artifacts capture the selection bonus and performance degrades as the bonus grows, isolating the wall\-aware entrance test as the component that makes the method work—and a preliminary positive on HM3D scanned buildings, where the speedup over Go\-Explore tracks scene difficulty \(r=0\.69r\{=\}0\.69\) even as frontier selection dominates blanket coverage\. The evidence supports a deliberately scoped claim: topology\-aware selection pays off where enclosed structure must be discriminated, and remains competitive at open coverage—where frontier methods are strongest—despite not being tuned for that regime\.

## 1Introduction

Hard\-exploration methods answer the question*“where should the agent go next?”*with proxies: count\-based and curiosity methods prefer rarely\-visited or poorly\-predicted states\[[2](https://arxiv.org/html/2607.09971#bib.bib2),[15](https://arxiv.org/html/2607.09971#bib.bib15),[4](https://arxiv.org/html/2607.09971#bib.bib4)\]; Go\-Explore\[[10](https://arxiv.org/html/2607.09971#bib.bib10)\]archives visited cells and returns to under\-visited ones; frontier exploration\[[19](https://arxiv.org/html/2607.09971#bib.bib19)\]drives to the boundary between known and unknown space\. All of these treat the unexplored region as undifferentiated\. Real environments are not undifferentiated: buildings contain rooms behind doorways, debris fields contain cavities, and—crucially—many enclosed regions are*sealed*: walled pockets, closed containers, reconstruction artifacts\. An explorer that cannot tell an enterable enclosure from a sealed one wastes budget on boundaries that can never pay off, and dilutes its attention across large rims when only a single gap cell actually admits entry\.

This failure mode matters beyond benchmark exploration: navigation planners consume maps whose useful content is precisely paths, obstructions, and rooms, and an exploration mechanism that explicitly discovers*which enclosed regions exist, which are enterable, and where their entrances are*produces exactly that structure as a by\-product of exploring\.

TopoExplore makes this distinction explicit\. It keeps the Go\-Explore outer loop \(select an archived cell, return to it, explore randomly, record new cells\) and adds a periodic*topological pass*: enclosed unexplored regions of the visited\-set occupancy grid are detected, and a selection bonus is placed on the cells from which an action actually enters them—their*strict entrances*—weighted by unexplored area and decayed by the number of inward actions already tried\. This construction has three consequences\. A sealed region has no entrance, so it never receives a bonus\. A large rim cannot dilute the signal, because only the few cells that actually admit entry qualify\. And once a region has been entered, or its entrances exhausted, it retires and is never targeted again\.

#### Contributions\.

1. 1\.To our knowledge, the first method in which the topology of the*visited set*—its persistent homology—decides*where to explore next*\. Robotics has long used topology for navigation, but on the other side of the problem: topological maps and Voronoi graphs organize already\-explored space, and homology\-aware planners select*how*to reach a goal among path classes\[[9](https://arxiv.org/html/2607.09971#bib.bib9),[3](https://arxiv.org/html/2607.09971#bib.bib3)\]; RL uses topology descriptively \(§[6](https://arxiv.org/html/2607.09971#S6)\)\. Turning homology into the exploration decision engine itself opens what we believe is a promising direction for exploration research\.
2. 2\.A topology\-aware selection rule for archive\-based exploration, exact for binary occupancy grids and cheap \(scipy flood fill; no TDA dependency at run time\), with a strict\-entrance test and a retirement mechanism \(§[2](https://arxiv.org/html/2607.09971#S2)\)\.
3. 3\.A controlled 18\-environment suite that varies decoy count, chamber geometry, nesting, and door difficulty independently, with ground\-truth topology, against six baselines including a frontier method on the identical archive/return harness \(§[3](https://arxiv.org/html/2607.09971#S3)\)\.
4. 4\.An honestly reported negative \(Montezuma’s Revenge\) with a bit\-exact instrumented autopsy identifying*why*the mechanism fails without wall knowledge \(§[4](https://arxiv.org/html/2607.09971#S4)\), and a preliminary positive on HM3D scanned buildings \(§[5](https://arxiv.org/html/2607.09971#S5)\)\.
5. 5\.A scoping result we believe the field should adopt when evaluating structured exploration: with free archive returns, frontier selection is near\-optimal for*blanket coverage*\(2\.58×\\timesover Go\-Explore on HM3D\)—the interesting regime for topology\-aware methods is*discrimination under sealed structure*, where frontier degrades and TopoExplore does not\.

## 2Method

### 2\.1Setting and outer loop

We adopt the Go\-Explore exploration phase\[[10](https://arxiv.org/html/2607.09971#bib.bib10)\]: an archive maps discretized cells to snapshots; each iteration selects a cell with probability proportional to a score, restores its snapshot, and explores randomly for a fixed horizon, adding newly seen cells\. Our cells are positions on a grid \(MiniGrid;[8](https://arxiv.org/html/2607.09971#bib.bib8)\), RAM\-position tiles \(Atari;[1](https://arxiv.org/html/2607.09971#bib.bib1)\), or\(f​l​o​o​r,0\.5​m×0\.5​m\)\(floor,0\.5\\text\{m\}\\times 0\.5\\text\{m\}\)pose bins \(Habitat;[18](https://arxiv.org/html/2607.09971#bib.bib18)\); observations are never used for the archive key\. The selection score is the Go\-Explore CellScore with one added term:

Score​\(c\)\\displaystyle\\mathrm\{Score\}\(c\)=∑aCnt​\(c,a\)\+∑nNeigh​\(c,n\)\+Topo​\(c\)\+1,\\displaystyle=\\textstyle\\sum\_\{a\}\\mathrm\{Cnt\}\(c,a\)\+\\sum\_\{n\}\\mathrm\{Neigh\}\(c,n\)\\;\+\\;\\mathrm\{Topo\}\(c\)\\;\+\\;1,\(1\)Cnt​\(c,a\)\\displaystyle\\mathrm\{Cnt\}\(c,a\)=wa​\(1/\(v​\(c,a\)\+ε1\)\)1/2\+ε2,Topo​\(c\)=mag​\(Vc\)​wtopo​\(1/\(nin​\(c\)\+ε1\)\)1/2,\\displaystyle=w\_\{a\}\\,\(1/\(v\(c,a\)\+\\varepsilon\_\{1\}\)\)^\{1/2\}\+\\varepsilon\_\{2\},\\qquad\\mathrm\{Topo\}\(c\)=\\mathrm\{mag\}\(V\_\{c\}\)\\,w\_\{\\mathrm\{topo\}\}\\,\(1/\(n\_\{\\mathrm\{in\}\}\(c\)\+\\varepsilon\_\{1\}\)\)^\{1/2\},\(2\)whereTopo​\(c\)\\mathrm\{Topo\}\(c\)is present only whenccis a strict entrance of a live voidVcV\_\{c\},mag\\mathrm\{mag\}is the void’s unexplored free\-interior area, andnin​\(c\)n\_\{\\mathrm\{in\}\}\(c\)counts inward actions already executed fromcc\. Settingwtopo=0w\_\{\\mathrm\{topo\}\}=0recovers Go\-Explore*exactly*\(same code path\): every comparison below is a strict one\-term ablation\.

### 2\.2Topological pass

LetMMbe the boolean visited\-cell mask\. A*void*is a connected component of¬M\\lnot Mthat does not touch the grid border\. For binary masks this flood\-fill definition is not an approximation: treating each visited cell as a filled square glued to its neighbours turnsMMinto a shape \(its*cubical complex*\), and the enclosed unvisited components are exactly that shape’sH1H\_\{1\}classes—its holes; we verify agreement against a GUDHI cubical\-persistence implementation\[[13](https://arxiv.org/html/2607.09971#bib.bib13)\]in unit tests, and run the scipy flood fill in all experiments\. Each void records its interior cells, its rim \(the visited cells bordering the interior\), and its area\. Voids are re\-detected from scratch on every pass, so we match them to the previous pass by interior overlap \(IoU\); a void therefore keeps a stable identity as the map grows, and once retired it stays retired\.

### 2\.3Strict entrances and retirement

On a large rim, only cells from which an action*actually enters*the void receive the bonus: cellccis a strict entrance iff it is 4\-adjacent to an unvisited*free*\(non\-wall\) interior cell, or adjacent to a door cell of the void\. Wall\-facing actions never qualify\. Consequently \(a\) sealed voids \(solid ring, no door\) have no entrance and are never targeted; \(b\) the bonus concentrates on gap cells rather than diluting over the rim; \(c\) theninn\_\{\\mathrm\{in\}\}decay retires an entrance as its inward actions are consumed\. The wall/door knowledge comes from the environment’s occupancy ground truth in MiniGrid, and from the navigation mesh in Habitat; it is genuinely unavailable in Atari RAM—§[4](https://arxiv.org/html/2607.09971#S4)shows this is exactly the component whose absence breaks the method, which we regard as evidence the mechanism \(and not some side effect\) drives the gains\.

## 3Controlled suite \(MiniGrid\)

![Refer to caption](https://arxiv.org/html/2607.09971v1/figures/prelim_minigrid_panel.png)Figure 1:The 18 main\-evaluation environments: one enterable chamber \(green, gap or door entrance\) among sealed decoys \(orange, solid ring\), varying decoy count, chamber shape, nesting depth, and door interaction count\.#### Protocol\.

The TopoExplore preset \(wtopo=100w\_\{\\mathrm\{topo\}\}\{=\}100, area magnitude, strict entrances\) was selected on held\-out sweeps and then*frozen*; all numbers below use it unchanged\. 15 seeds, 1\.2M\-step budget; metric: median steps to first entry of the enterable chamber\. Baselines share the recorder; Go\-Explore, frontier \(uniform over frontier cells\) and nearest\-frontier share the identical archive/return harness and differ only in selection; count\-bonus and RND\-greedy are signal\-only controllers; PPO\+RND and PPO\+ICM are learned policies with intrinsic reward \(Appendix[A](https://arxiv.org/html/2607.09971#A1)\)\.

Table 1:Speedup of median steps\-to\-first\-entry vs\. the Go\-Explore ablation \(\>1\>1favors the method; “—” = never entered within budget; 15 seeds\)\.Boldmarks the best archive/return method per row\. Frontier*degrades when decoys appear*\(Decoys1–5\) and on hard doors \(Tries3–5\); TopoExplore holds both regimes\.
#### Findings\.

\(Table[1](https://arxiv.org/html/2607.09971#S3.T1)\.\)\(1\) Discrimination is the win condition\.On every environment with≥\\geq1 sealed decoy, TopoExplore beats frontier \(2\.11 vs\. 0\.83 at one decoy; 1\.89 vs\. 1\.32 at five\); on the decoy\-free control, frontier wins \(1\.47 vs\. 1\.18\)\. The mechanism is visible in the frontier baseline itself: unvisited wall cells are indistinguishable from unexplored space, so sealed rims are*permanent frontier*that never resolves\.\(2\) Entrance persistence matters on hard doors\.A door requiring three interactions yields TopoExplore’s largest win over frontier \(10\.9 vs\. 2\.1\); signal\-only baselines never enter\.\(3\) Learned policies own sequential bottlenecks\.PPO\+RND dominates nested, concentric\-door layouts \(up to 4\.4×\\times\) — persistence of a learned policy, not selection, is the right tool there; we report this rather than tune it away\.\(4\) The advantage is geometry\-sensitive\.A circular chamber \(poor gap/rim ratio at this resolution\) is a genuine loss \(0\.55\); we discuss resolution coupling in §[7](https://arxiv.org/html/2607.09971#S7)\.

## 4An honest negative: Montezuma’s Revenge

![Refer to caption](https://arxiv.org/html/2607.09971v1/figures/prelim_atari_panel.png)Figure 2:Montezuma pipeline: ALE frame; the domain\-general downscaled archive key\[[10](https://arxiv.org/html/2607.09971#bib.bib10)\]; per\-room RAM\-tile occupancy grids on which the topological pass runs\. Note the enclosed unvisited pockets: some are real unexplored space, others are unreachable screen artifacts—and RAM provides no wall mask to tell them apart\.We ported the pass to Go\-Explore\-Atari \(downscaled\-image archive keys, ALE snapshot restore\) by running flood fill on per\-room RAM\-position occupancy grids\. The strict\-entrance test*cannot*be ported: RAM gives no wall mask, so the whole void rim was nominated, with decay by selection count\. We report the full weight sweep at 30M frames \(seed 0; the archive loop is bit\-exactly reproducible, verified by instrumented re\-runs\):

Go\-Explorew=0\.3w\{=\}0\.3w=1w\{=\}1w=10w\{=\}10w=100w\{=\}100PPO\+RNDPPO\+ICMrooms found19181316811mean per\-room coverage0\.920\.900\.750\.940\.61——
No weight beats the ablation; damage grows with weight\. The tile\-level autopsy shows why: every run’s occupancy contains∼\\sim5–6 voids that*never*resolve—unreachable screen artifacts—and atw=100w\{=\}100these captured 94% of selections as permanent attractors\. The failure is not that voids are absent or that they do not resolve \(Go\-Explore resolved 10/15 of its live voids by 10M frames\); it is that*unresolvable*voids cannot be retired without wall knowledge\. This isolates the strict\-entrance test as the load\-bearing component and predicted the design of §[5](https://arxiv.org/html/2607.09971#S5)\.

## 5Preliminary: HM3D scanned buildings

![Refer to caption](https://arxiv.org/html/2607.09971v1/figures/prelim_hm3d_panel.png)Figure 3:HM3D minival scenes as the runner rasterizes them \(main floor band of the navigation\-mesh mask\)\. Left pair: small scans a random walk saturates, where all methods tie; right pair: complex multi\-room scans where TopoExplore wins \(1\.28×\\times, 1\.40×\\times\)\.In habitat\-sim, on HM3D scanned buildings\[[16](https://arxiv.org/html/2607.09971#bib.bib16)\], we recover the missing ingredient: the navigation mesh is a ground\-truth wall mask, so strict entrances port \(a rim cell qualifies iff adjacent to a*navigable*unvisited void cell\), and sealed reconstruction artifacts are never targeted\. On the 10\-scene minival split \(10 seeds, 400k steps, spawn and metrics restricted to the largest navmesh island\): TopoExplore reaches 95% coverage 1\.12×\\timesfaster than Go\-Explore \(geometric mean\), the per\-scene speedup tracks how hard the scene is to saturate \(r=0\.69r\{=\}0\.69against1−AUCG​o−E​x​p​l​o​r​e1\-\\mathrm\{AUC\}\_\{Go\-Explore\{\}\}; largest win 1\.40×\\timeson the hardest scene\), and the selection trace is surgical \(1–2% of selections topo\-influenced; entrances retire as rooms are entered\)\.

#### Scoping result: frontier owns blanket coverage\.

A frontier baseline on the same harness \(uniform over frontier cells, given the same navmesh information\) reaches 95% coverage 2\.58×\\timesfaster than Go\-Explore—beating both Go\-Explore and TopoExplore on every scene\. With free archive returns and known navigability, “always jump to the boundary of the unknown” is near\-optimal greedy*coverage*, and blanket coverage contains no discrimination problem: HM3D scans have few large sealed enclosures reachable\-looking enough to fool a navmesh\-informed frontier\. We therefore claim nothing about open coverage\. The regime where topology\-aware selection earns its complexity is the one Table[1](https://arxiv.org/html/2607.09971#S3.T1)isolates—sealed structure that boundary methods cannot tell apart from opportunity—and the follow\-up evaluation \(held\-out 100\-scene split; region\-entry and waste\-on\-sealed metrics; sensor\-limited frontier without ground\-truth navigability; travel\-costed returns\) is designed to test exactly that at scale\.

## 6Related work

Archive\-based exploration\.Go\-Explore\[[10](https://arxiv.org/html/2607.09971#bib.bib10)\]introduced return\-then\-explore with rarity\-weighted cell selection; policy\-based variants followed\[[11](https://arxiv.org/html/2607.09971#bib.bib11)\]\. TopoExplore changes only the selection rule\.Intrinsic motivation\.Count\-based bonuses\[[2](https://arxiv.org/html/2607.09971#bib.bib2)\], prediction\-error curiosity\[[15](https://arxiv.org/html/2607.09971#bib.bib15)\], and random\-network distillation\[[4](https://arxiv.org/html/2607.09971#bib.bib4)\]supply exploration signals without structural knowledge of the unexplored; NovelD\[[20](https://arxiv.org/html/2607.09971#bib.bib20)\]shapes them with novelty differences\. These are complementary—our PPO baselines use them\.Frontier and active mapping\.Frontier exploration\[[19](https://arxiv.org/html/2607.09971#bib.bib19)\]and its modern planning descendants \(e\.g\. TARE\[[6](https://arxiv.org/html/2607.09971#bib.bib6)\]; GLEAM\[[7](https://arxiv.org/html/2607.09971#bib.bib7)\]for learned active mapping\) target the known/unknown boundary; they are the strongest baseline for coverage but undifferentiated w\.r\.t\. enclosure and sealedness\.Topology in robotics and RL\.Topology is well established in robotic navigation, but downstream of exploration: generalized Voronoi graphs give topological maps built during sensor\-based exploration\[[9](https://arxiv.org/html/2607.09971#bib.bib9)\], persistent homology selects among homotopy classes of*paths*through mapped space\[[3](https://arxiv.org/html/2607.09971#bib.bib3)\], and frontier rankings have used topological map abstractions—all deciding*how*to traverse or verify what is already known\. In learning, topological memory for navigation\[[17](https://arxiv.org/html/2607.09971#bib.bib17)\]and persistent\-homology analyses of learned representations\[[14](https://arxiv.org/html/2607.09971#bib.bib14)\]are descriptive\. To our knowledge, using the homology of the*visited set*to decide*where to explore next*—return\-target selection with entrance gating and retirement—is new\.

## 7Discussion and limitations

What the evidence supports\.Topology\-aware selection pays off when \(a\) enclosed structure is real and observable in the occupancy geometry, \(b\) sealed decoys exist to discriminate, and \(c\) a wall/traversability signal distinguishes gaps from walls\. Remove \(c\) and the method degrades monotonically with its own weight \(Montezuma\); remove \(b\) and a frontier method is the better tool \(Decoys0; HM3D blanket coverage\)\.Limitations\.Single\-seed Atari sweep; minival\-only Habitat preview \(the held\-out evaluation is in progress\); grid\-resolution coupling \(the circle loss\); free\-teleport cost model flatters all archive methods, frontier most; the frontier baseline here is selection\-level, not a full planning stack\.Outlook: beyond grids\.Everything here leans on the map being a grid: gluing visited cells together edge\-to\-edge produces a shape whose enclosed holes are exactly what our flood fill detects\. Where no natural grid exists, that shape must be built directly from sampled states: Rips complexes connect states that are close under a distance function, and*Dowker complexes*do the same under a*quasimetric*—an asymmetric distance—which matters because action reachability is asymmetric: a region that is cheap to enter but expensive to exit is invisible to any symmetric construction\. These tools also lift the mechanism beyondH1H\_\{1\}and beyond physical space; we develop this direction in follow\-up work\.v2\.The expanded version will add the held\-out HM3D evaluation with region\-entry/waste metrics and sensor\-limited frontier, IQM/CI statistics, NovelD, and a manipulation suite with sealed\-container discrimination\.

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## Appendix AExperimental details

#### Frozen constants\.

wtopo=100w\_\{\\mathrm\{topo\}\}\{=\}100,ptopo=0\.5p\_\{\\mathrm\{topo\}\}\{=\}0\.5, area magnitude, strict entrances; Go\-Explore weightswseen=0\.3w\_\{\\mathrm\{seen\}\}\{=\}0\.3,whoriz=0\.3w\_\{\\mathrm\{horiz\}\}\{=\}0\.3,wvert=0\.1w\_\{\\mathrm\{vert\}\}\{=\}0\.1,p=0\.5p\{=\}0\.5,ε1=10−3\\varepsilon\_\{1\}\{=\}10^\{\-3\},ε2=10−5\\varepsilon\_\{2\}\{=\}10^\{\-5\}\(grid\-searched once on a held\-out environment, then frozen; the functional form follows[10](https://arxiv.org/html/2607.09971#bib.bib10)\)\.

#### Suite \(§[3](https://arxiv.org/html/2607.09971#S3)\)\.

41×\\times41 grids; archive loop: deepcopy snapshot restore, uniform random explore bursts of 30 steps, topological pass every 2,000 steps; 15 seeds×\\times8 methods×\\times18 environments \(2,160 runs\)\. Frontier baselines run the identical loop with selection replaced by uniform\-over\-frontier \(or BFS\-nearest frontier, which collapses to 0\.67×\\timesunder free returns and is reported as an ablation\)\. PPO: 2×\\times128 tanh actor\-critic, lr2\.5×10−42\.5\{\\times\}10^\{\-4\},γ=0\.99\\gamma\{=\}0\.99, clip 0\.2; RND 128→\\to64 MLPs; ICM inverse\-trained 64\-d features \(β=0\.2\\beta\{=\}0\.2\)\. ICM’s intrinsic reward collapses in deterministic symbolic gridworlds \(forward model masters the dynamics; measured 15×\\timesdecay on Atari within 5M frames\), consistent with large\-scale studies of curiosity\-driven learning\[[5](https://arxiv.org/html/2607.09971#bib.bib5)\]—a faithful implementation, structurally disadvantaged in this domain\.

#### Montezuma \(§[4](https://arxiv.org/html/2607.09971#S4)\)\.

Go\-Explore\-Atari: 11×\\times8×\\times8 downscaled\-image archive key, ALEcloneSystemStaterestore \(environment sticky actions off for exact restore, cf\.[12](https://arxiv.org/html/2607.09971#bib.bib12)\), sticky\-action random explore \(repeat 0\.95, horizon 100\); topological pass on per\-room RAM\-\(x,y\)/8\(x,y\)/8tiles every 50 iterations; 30M frames\. Determinism verified by bit\-identical instrumented re\-runs; per\-room coverage computed against the union of tiles reached by any run\. PPO baselines: Nature\-CNN, 8 vectorized envs, 15–16M frames\.

#### HM3D \(§[5](https://arxiv.org/html/2607.09971#S5)\)\.

habitat\-sim 0\.3\.3, discrete actions \(0\.25m forward, 30∘turns\), teleport returns, sticky random\-walk bursts \(repeat 0\.8, horizon 50\); cell==\(floor band, 0\.5m2\); navigation\-mesh wall mask by direct pathfinder sampling; spawn and denominators restricted to the largest navmesh island \(HM3D meshes fragment; up to 60% of nominally navigable area is unreachable\)\. Minival split only in this version; the held\-out 100\-scene evaluation is reserved for v2\.

#### Reproducibility\.

All archive\-loop results are deterministic given \(seed, config\); a verification script checks regenerated artifacts against frozen expected values \(exact for CPU loops, tolerance\-based for GPU/simulator paths\)\. Code, environment suite, and expected\-results manifests will accompany v2\.