✗ Failed on benchmark 2026

Differentiable Persistence Landscape Layer

Implementation & benchmark of arXiv:2608.08858 — A Hilbert space embedding of persistence diagrams and barcodes

Usefulness7/10

Difficulty4/10

Novelty5/10

Architecture Stability Accuracy Sample efficiency Graph nn Embedding Loss Topology Functional analysis Metric geometry Optimal transport

Source paper: A Hilbert space embedding of persistence diagrams and barcodes arXiv:2608.08858 ⓘ · analyzed Sep 1, 2026

AI-generated research hypothesis, automatically tested. Not peer-reviewed.

Idea description

Convert each persistence diagram produced from an input, intermediate feature map, or graph filtration into a discretized persistence landscape and feed it to an MLP or concatenate it with ordinary neural features. Unlike a variable-size list of birth-death pairs, the landscape has a fixed tensor shape and is provably nonexpansive with respect to the diagram Wasserstein distance.

Formulas

$$\triangle_{(b,d)}(t)=\max\{0,\min(t-b,d-t)\}$$

$$\Lambda_\alpha(k,t)=\operatorname{kmax}_{j=1}^{n}\bigl(\triangle_{x_j}(t)\bigr)$$

$$\|\Lambda_\alpha-\Lambda_\beta\|_{L^p(\mathbb{N}\times\mathbb{R})}\le W_p^\triangle(\alpha,\beta)$$

$$\widehat{\Lambda}_\alpha[k,r]=\operatorname{kmax}_{j}\left(\max\{0,\min(t_r-b_j,d_j-t_r)\}\right),\qquad t_r=t_{\min}+r\Delta t$$

Mathematical statement

For a point or interval $x=(b,d)$ with $b<d$, define its tent function by $\triangle_x(t)=\max\{0,\min(t-b,d-t)\}$. For a finite diagram $\alpha=\sum_{j=1}^{n}x_j$, Definition 2.6 defines the persistence landscape by $\Lambda_\alpha(k,t)=\operatorname{kmax}_{j=1}^{n}(\triangle_{x_j}(t))$, where $\operatorname{kmax}$ is the $k$-th largest value and missing values are zero. Theorem 4.6 states that for $1\le p\le\infty$, $\|\Lambda_\alpha-\Lambda_\beta\|_{L^p(\mathbb{N}\times\mathbb{R})}\le W_p^\triangle(\alpha,\beta)$, where $W_p^\triangle$ is the $p$-Wasserstein matching distance using the diagonal cost $d_q(x,y)=\|\triangle_x-\triangle_y\|_q$. For $p=2$, discretizing $t$ gives a Euclidean Hilbert feature vector. The inequality means diagram perturbations cannot be amplified by the landscape map.

Implementation notes

Integrate this after a differentiable or frozen topological preprocessing step that produces a persistence diagram per example, graph, image, or intermediate feature channel. The tensor input is a padded diagram $\{(b_j,d_j)\}_{j=1}^{N}$, with points of persistence below a threshold optionally removed. Choose a fixed grid $t_r=t_{\min}+r\Delta t$, retain the first $K$ landscape levels, and mask padded points. Compute v[j,r] = max(0, min(t[r]-birth[j], death[j]-t[r])), sort the $N$ values along the point axis, and set L[k,r] to the $k$-th largest value, padding nonexistent levels with zero. Flatten or average-pool L, normalize by the grid measure, and concatenate it with the baseline neural representation before the prediction head. In pseudocode: v = relu(min(t[None,:]-birth[:,None], death[:,None]-t[None,:])); L = topk(v, K, dim=points); z = concat(backbone(x), flatten(L)). The paper supplies the stability guarantee; estimate only grid truncation error, persistence threshold effects, and finite-$K$ truncation empirically. Start with MUTAG or a small CIFAR-10 filtration model, comparing landscapes against persistence images, padded birth-death coordinates, and a diagram-distance baseline. Measure clean accuracy and degradation under synthetic birth/death noise, diagram cardinality changes, and input perturbations. Success is equal or better clean accuracy with smoother predictions and lower accuracy degradation as perturbation magnitude increases.

Verification

Failed on benchmark

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Not reproduced ✗

Stage 2 · Practical benchmark result: Inconclusive

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 9/10

Built a differentiable PyTorch persistence-landscape layer with masking, fixed grid, top-K levels, flattening, and a NumPy reference implementation. The mechanism manifested: single-point landscape distance divided by matched tent distance was 0.9995–0.99999, the 100 multi-point trials had maximum ratio 0.99970 and mean 0.90268, and grid integration error decreased from 3.46e-4 at dt=0.02 to 2.98e-6 at dt=0.00125. The toy classifier showed no accuracy win: both padded birth-death coordinates and landscapes achieved 1.0 clean and noisy accuracy.

Agent confidence: 9/10

Baseline: Padded birth-death MLP: 1.0 clean accuracy and 1.0 at noise sigma 0.01, 0.03, and 0.07.
Idea: Landscape MLP: 1.0 clean accuracy and 1.0 at noise sigma 0.01, 0.03, and 0.07. Stability checks: single-point ratios 0.9995–0.99999; multi-point maximum ratio 0.99970 and mean 0.90268; finite nonzero autograd gradients; grid error 3.46e-4 to 2.98e-6 as dt decreases.

Limitations: The benchmark uses synthetic equal-cardinality diagrams and a tiny MLP, not MUTAG, CIFAR-10, or diagrams generated by an actual persistence algorithm. It does not compare persistence images, input perturbations, varying cardinality with real padding masks, diagonal matching, finite-K truncation error, or end-to-end differentiable topological preprocessing.

How to run: python3 landscape_experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2608.08858) Sep 2, 2026

No significant difference vs. the tuned baseline (+200.0%, p=0.1196, wins 0 / 8 paired seeds).

Benchmark: Persistence_diagrams
Model: mlp_tiny
Paired seeds: 8
Baseline mean: 0.0012
Idea mean: 0.0037
Effect (Δ): 0.0025 (+200.0%; negative = idea better)
Wins: 0 / 8 paired seeds
p-value: 0.1196 (permutation test, 20 000 shuffles)
Smallest detectable effect: n/a
Mechanism: Not confirmed ✗
Practical effect: Inconclusive
Baseline tuning: swept over 3 configs

Limitations:

The registered custom task uses synthetic diagrams rather than diagrams generated from real images or graphs. Persistence images, diagonal matching, larger backbones, input perturbations, and end-to-end differentiable topological preprocessing were not tested. The mechanism signature is a direct layer probe rather than valid trained-model behavioral evidence.

How to run: python3 stage2_bench.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Sep 1, 2026 latest Failed ✗

Agent confidence: 10/10

Implemented and benchmarked a differentiable persistence landscape layer in a separately trained MLP system on the registered persistence_diagrams custom track. The baseline achieved lower test error than the landscape system, with paired delta +0.00250 and permutation p=0.11955, so there was no significant win. The observed stability-probe ratio was 1.342 versus the claimed bound 1.0, so that mechanism check was not confirmed.

Baseline: Best lr=0.003; full 8-seed mean error 0.00125, std 0.00217; sweep means: lr=0.001 -> 0.03625, lr=0.003 -> 0.00250, lr=0.01 -> 0.00250.
Idea: Best lr=0.001; 8-seed mean error 0.00375, std 0.00331; per-seed errors [0.0,0.005,0.005,0.01,0.0,0.005,0.0,0.005].

Benchmark result