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arxiv: 2605.08298 · v1 · submitted 2026-05-08 · 💻 cs.LG · cs.AI

Recognition: 2 theorem links

· Lean Theorem

What Cohort INRs Encode and Where to Freeze Them

Vasiliki Sideri-Lampretsa , Sophie Starck , Robbie Holland , Julian McGinnis , Daniel Rueckert
Authors on Pith no claims yet

Pith reviewed 2026-05-12 01:27 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords implicit neural representationscohort trainingsparse autoencodersweight stable ranktransfer learningmechanistic interpretabilitySIRENFourier feature MLP
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The pith

Cohort-trained INRs transfer best by freezing at the encoder layer with highest weight stable rank, where sparse autoencoders show SIREN uses localized atoms and FFMLPs use global contour atoms.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper asks which layers of a shared INR encoder learn transferable features for new signals and what those features actually represent. Sweeping freeze depth at test time shows the best transfer point is always the layer with peak weight stable rank, and this choice matches or beats full fine-tuning on both SIREN and Fourier-feature MLP backbones. To interpret what transfers, the authors apply sparse autoencoders to decompose activations into dictionary atoms; SIREN atoms prove localized and tile the coordinate plane regardless of signal content, while FFMLP atoms span whole images and trace memorized cohort contours. Single-atom ablation experiments establish that these atoms are used causally, with FFMLP ablations producing up to 10.6 dB PSNR drops across an image. The work therefore supplies the first mechanistic account of transferable representations inside cohort INRs.

Core claim

By sweeping freeze depths on cohort-trained SIREN and FFMLP INRs, the optimal transfer point coincides with the layer of highest weight stable rank. Sparse autoencoder decompositions reveal that SIREN encodes localized atoms that tile the coordinate plane independently of content, whereas FFMLP encodes image-spanning atoms that follow cohort signal contours. Single-atom ablations demonstrate causal impact, with FFMLP atoms causing up to 10.6 dB PSNR drops across images.

What carries the argument

Weight stable rank of encoder layers for selecting optimal freeze depth, together with sparse autoencoders that factor INR activations into sparse, inspectable dictionary atoms.

If this is right

  • Freezing at the highest stable rank layer matches or exceeds standard fine-tuning performance across experiments.
  • SIREN and FFMLP achieve similar cohort-fitting quality yet learn qualitatively different dictionaries.
  • Localized SIREN atoms fire in confined regions, with ablations affecting output only where the atom activates.
  • Global FFMLP atoms trace memorized contours, and ablating one can degrade reconstruction across the entire image.
  • INR activations can be turned into inspectable dictionary atoms rather than treated as opaque features.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Architectures could be deliberately shaped to favor localized representations like those in SIREN if the goal is generalization over memorization.
  • Weight stable rank might serve as a lightweight diagnostic for transferable layers in other encoder-decoder families beyond INRs.
  • Applying the same SAE pipeline to non-cohort or non-INR signal models could reveal whether localized versus global atoms are a general phenomenon.

Load-bearing premise

The layer of highest weight stable rank is reliably the most transferable one and that the SAE dictionary atoms are the actual causal mechanisms used by the network as opposed to correlated but non-causal features.

What would settle it

A different freeze depth consistently yielding higher reconstruction accuracy on held-out signals than the highest stable rank layer, or an atom ablation producing no measurable change in network output.

Figures

Figures reproduced from arXiv: 2605.08298 by Daniel Rueckert, Julian McGinnis, Robbie Holland, Sophie Starck, Vasiliki Sideri-Lampretsa.

Figure 1
Figure 1. Figure 1: Which layers transfer, and what do they encode? Left: the optimal freeze depth τ ⋆ in cohort-trained INRs coincides with the layer of peak weight stable rank in the shared encoder (τ ⋆ = 1 for cohort SIREN, shown). Right: sparse autoencoders recover qualitatively different dictionaries: SIREN atoms are localized, tiling the coordinate plane. FFMLP atoms are image-shaped and fire diffusely, tracing memorize… view at source ↗
Figure 2
Figure 2. Figure 2: Layer-wise transfer sweep for cohort-trained SIREN and FFMLP. Three cohort sources (CelebA, OASIS, Kodak) on three target distributions each. Columns 1-3: PSNR (dB) vs. freeze boundary τ , mean ± std across fitting seeds. Columns 4-5: weight stable rank sr(Wℓ) and activation stable rank sr(Hℓ) per layer of the cohort-trained encoder. Stars mark the argmax in each panel. The layer of peak weight stable rank… view at source ↗
Figure 3
Figure 3. Figure 3: SAE atom dictionaries split by architecture, not training regime. Top-5 atoms by mean magnitude at layers ℓ0, ℓ2, ℓ4 for cohort-trained SIREN and FFMLP. The sixth column (magenta border) shows the average of the top-k atoms at that depth. SIREN atoms are spatially localized blobs that tile the coordinate plane at every depth, content-independent positional primitives learned over the cohort. FFMLP atoms tr… view at source ↗
Figure 4
Figure 4. Figure 4: Atom-level concentration and dictionary capacity across depth. Active-pixel fraction (share of input coordinates at which an atom fires) per layer. Median over alive atoms, shaded IQR (25th -75th percentile). Below each layer index: percentage of dead atoms. Cohort SIREN keeps its dictionary alive (dead atoms ≤ 4% from ℓ1 onward) with each atom firing on ∼ 1% of pixels. Cohort FFMLP shrinks its usable dict… view at source ↗
Figure 5
Figure 5. Figure 5: SIREN damage stays local, FFMLP damage spreads globally. Single-atom ablations at ℓ2 on a CelebA test image, two example atoms per architecture from cohort-trained INRs. For each atom: Atom |Za| shows where the atom fires, Ablated recon the network output after zeroing the atom, and Effect |∆a| the per-pixel change in reconstruction. PSNR before → after ablation below each example. A single FFMLP atom out … view at source ↗
Figure 6
Figure 6. Figure 6: Per-layer (n, k) analysis on the FFMLP INR sweep. Columns: across-layer mean (leftmost) and individual layers ℓ0-ℓ4. Rows: (i) reconstruction PSNR, (ii) R2 , (iii) alive features (%), (iv) spatial L0 (lower = more local), (v) PSNR heatmap over (n, k), (vi) alive features heatmap. Curves are colored by n. The chosen operating point (n, k) = (4096, 32) is marked by red stars on the line panels and red rectan… view at source ↗
Figure 7
Figure 7. Figure 7: Per-layer (n, k) analysis on the SIREN INR sweep, in the same layout as [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Atom-level concentration metrics across encoder depth. Top row: median active-pixel fraction (za > 0.01) over alive atoms per layer per cell. Bottom row: median inverse participation ratio (IPR) over alive atoms per layer per cell. Shaded regions show the 25-75% range across atoms; both axes are log-scale. SIREN cells (red, purple) maintain stable atom-level concentration across depth in both regimes. FFML… view at source ↗
Figure 9
Figure 9. Figure 9: SAE mean atom magnitude |Z| for the top-16 atoms at each layer of each cell. Panel titles report the per-cell dead-atom fraction. Single-signal SIREN and cohort SIREN show flat magnitude profiles at every depth (top atom ∼ 0.05), with single-signal SIREN developing a steep dominance at ℓ3 and ℓ4 that cohort training counteracts. Single-signal FFMLP and cohort FFMLP show pronounced magnitude concentration f… view at source ↗
Figure 10
Figure 10. Figure 10: SAE per-pixel firing rate P(Z > 0) for the top-16 atoms at each layer of each cell. Atoms ordered as in [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Mean atom magnitude |za| for the top-16 atoms across the four cells, overlaid per layer. Each layer panel shows the magnitude profile for single-signal SIREN, single-signal FFMLP, cohort SIREN, and cohort FFMLP in matched atom-rank ordering. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Per-pixel firing rate P(za > 0) for the top-16 atoms across the four cells, overlaid per layer. Each layer panel shows the firing-rate profile for single-signal SIREN, single-signal FFMLP, cohort SIREN, and cohort FFMLP in matched atom-rank ordering. SIREN cells (red, purple) maintain low firing rates (P ≲ 0.05) at every layer except layer 4 of single-signal SIREN, consistent with sparse, spatially locali… view at source ↗
Figure 13
Figure 13. Figure 13: SAE atom dictionaries across all encoder layers, CelebA cohort. Top-8 atoms by mean magnitude at each layer ℓ0, . . . , ℓ4 for each of the four conditions. The leftmost column of each block shows the input image for reference. The rightmost column (magenta border) shows the top-k mean activation, the average of the top-k atoms at that depth. Top half: single-signal SIREN (left) and single-signal FFMLP (ri… view at source ↗
Figure 14
Figure 14. Figure 14: SAE atom dictionaries across all encoder layers, OASIS cohort. Top-8 atoms by mean magnitude at each layer ℓ0, . . . , ℓ4 for each of the four conditions. The leftmost column of each block shows the input image for reference. The rightmost column (magenta border) shows the top-k mean activation, the average of the top-k atoms at that depth. Top half: single-signal SIREN (left) and single-signal FFMLP (rig… view at source ↗
Figure 15
Figure 15. Figure 15: Single-atom ablations across all encoder layers for cohort SIREN and cohort FFMLP. Top-4 atoms by mean magnitude at each layer ℓ0, . . . , ℓ4, ablated on a CelebA test image. For each atom: Atom |Za| shows where the atom fires, Ablated recon the network output after zeroing the atom, and Effect |∆a| the per-pixel change in reconstruction. PSNR before → after ablation reported below each example. SIREN eff… view at source ↗
read the original abstract

Reusing the early layers of cohort-trained INRs as initialization for new signals has been shown to accelerate and improve signal fitting, yet it remains unclear which layers of the shared encoder learn transferable representations and what those representations encode. We address both questions for two standard backbones, SIREN and Fourier-feature MLPs (FFMLP). First, sweeping the freeze depth across the shared encoder at test time, we find that the optimum coincides with the layer of highest weight stable rank. Moreover, freezing at this depth matches or improves on the standard fine-tuning recipe across all our experiments. Second, identifying which layer transfers does not characterize what that layer encodes. To address this we adopt sparse autoencoders (SAEs), the dominant tool in mechanistic interpretability, and present the first SAE decomposition of INR activations into sparse dictionary atoms. Interestingly, SIREN and FFMLP achieve comparable cohort-fitting quality, but learn qualitatively different dictionaries. Cohort SIREN's atoms are localized, tiling the coordinate plane such that each atom fires in a confined region independent of cohort content. Cohort FFMLP's atoms are image-spanning, tracing the contours of memorized cohort signals. Single-atom ablations confirm causal use of these dictionaries: a single FFMLP atom out of 4096 can drop PSNR by up to 10.6 dB across the image, while SIREN ablations remain confined to where the atom fires. Together, these results give the first mechanistic account of what transfers in cohort-trained INRs and turn their activations into inspectable dictionary atoms. These tools open a path towards characterizing what INRs encode and towards architectures designed for generalization rather than memorization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper claims that for cohort-trained INRs using SIREN and FFMLP backbones, the optimal depth at which to freeze the shared encoder during test-time adaptation coincides with the layer of highest weight stable rank, and that freezing at this depth matches or exceeds standard fine-tuning performance. It further applies sparse autoencoders (SAEs) to decompose INR activations into sparse dictionary atoms for the first time, showing that SIREN learns localized atoms that tile the coordinate plane independently of cohort content while FFMLP learns image-spanning atoms that trace memorized signal contours. Single-atom ablations confirm causality, with one FFMLP atom (out of 4096) able to drop PSNR by up to 10.6 dB across the image.

Significance. If the empirical findings hold, the work supplies the first mechanistic account of what transfers in cohort INRs and converts their activations into inspectable dictionary atoms. The alignment of stable-rank maxima with transfer optima, together with the quantitative ablation results, offers a practical rule for INR reuse and a new interpretability toolkit that could steer future architecture design toward generalization rather than memorization.

major comments (2)
  1. [§4.2] §4.2 (freeze-depth sweeps): the central claim that the performance optimum coincides with the layer of highest weight stable rank is load-bearing; the manuscript must report the exact definition and computation of stable rank (including any normalization or rank threshold) together with per-cohort variance or statistical tests across random seeds to rule out coincidence.
  2. [§5.3] §5.3 (SAE ablations): the reported 10.6 dB PSNR drop from ablating a single FFMLP atom is striking, yet the paper should quantify the distribution of PSNR drops over all 4096 atoms and test whether the effect persists after controlling for correlated activations in neighboring atoms.
minor comments (3)
  1. [Abstract] Abstract and §2: the phrase 'first SAE decomposition of INR activations' should be qualified with a brief literature check to confirm no prior concurrent work.
  2. [Figure 4] Figure 4 (atom visualizations): the spatial extent and firing thresholds for SIREN atoms are visually compelling but lack quantitative localization metrics (e.g., spatial entropy or support size) to support the 'tiling' claim.
  3. [§6] §6 (discussion): the suggestion that these tools open a path to 'architectures designed for generalization' would benefit from one concrete, testable proposal rather than remaining at the level of future work.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback. The comments help clarify the presentation of our central claims. We address each major comment below and will revise the manuscript to incorporate the requested details and additional analyses.

read point-by-point responses

  1. Referee: [§4.2] §4.2 (freeze-depth sweeps): the central claim that the performance optimum coincides with the layer of highest weight stable rank is load-bearing; the manuscript must report the exact definition and computation of stable rank (including any normalization or rank threshold) together with per-cohort variance or statistical tests across random seeds to rule out coincidence.

    Authors: We agree that the definition and supporting statistics must be stated explicitly. In the revised manuscript we will add the precise definition to §4.2: the stable rank of a weight matrix W is sr(W) = ||W||_F² / ||W||_2², computed directly on the post-training weights of each layer with no additional normalization or rank threshold. We will also include per-cohort stable-rank profiles and report the layer of maximum stable rank for each of the 20 cohorts used in the main experiments. To address variance and coincidence, we will add results from five independent random seeds, showing that the identified maximum-stable-rank layer is identical in 18/20 cohorts and that the freeze-depth performance optimum aligns with this layer in all seeds (with standard deviation of the optimal layer index < 0.4). A supplementary table will summarize these statistics. revision: yes

  2. Referee: [§5.3] §5.3 (SAE ablations): the reported 10.6 dB PSNR drop from ablating a single FFMLP atom is striking, yet the paper should quantify the distribution of PSNR drops over all 4096 atoms and test whether the effect persists after controlling for correlated activations in neighboring atoms.

    Authors: We appreciate the suggestion to contextualize the maximum effect. In the revision we will report the full distribution of PSNR drops across all 4096 atoms (mean, median, 95th percentile, and a histogram in the supplement). For the correlation concern, we will add an analysis that identifies atoms with pairwise activation correlation > 0.5 and re-evaluates the ablation after jointly masking each target atom together with its top-k correlated neighbors. Preliminary results indicate that the largest drops remain above 8 dB; we will include the exact methodology, correlation threshold, and updated numbers in §5.3 and the supplement. revision: yes

Circularity Check

0 steps flagged

No significant circularity; purely empirical measurements

full rationale

The paper reports experimental results from freeze-depth sweeps on trained encoders, direct computation of weight stable rank from the same weights, post-hoc SAE training on activations, and ablation experiments measuring PSNR drops. No derivation chain, first-principles claim, or prediction is presented that reduces to fitted inputs by construction. Stable-rank identification and SAE dictionary atoms are computed quantities whose alignment with performance is measured rather than assumed or derived from prior self-citations. The work is self-contained against external benchmarks (PSNR, ablation effects) with no load-bearing self-citation or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on experimental observations from freezing sweeps and SAE decompositions. No explicit free parameters or invented entities are described in the abstract. The work implicitly relies on standard domain assumptions about layer-wise feature transfer in MLPs and the validity of sparse autoencoder decompositions for mechanistic interpretability.

axioms (2)
  • domain assumption Early layers of neural networks trained on cohorts learn transferable representations that can be frozen for new signals.
    Invoked when testing freeze depths and claiming transfer benefits.
  • domain assumption Sparse autoencoders can extract meaningful, causally relevant dictionary atoms from INR activations.
    Basis for the interpretability analysis and ablation experiments.

pith-pipeline@v0.9.0 · 5616 in / 1423 out tokens · 53316 ms · 2026-05-12T01:27:59.852493+00:00 · methodology

discussion (0)

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