A Deeper Analysis of Block-Sparse Featurizers

Alexandru-Iulius Jerpelea; Amith Ananthram

cs.LGAug 27, 2026

A Deeper Analysis of Block-Sparse Featurizers

Authors: Alexandru-Iulius Jerpelea, Amith Ananthram

Organizations: Columbia University

Abstract

The recently introduced block-sparse featurizer (BSF; Fel et al., 2026) is similar to a sparse autoencoder (SAE), but its atomic unit is a small subspace (a block of directions) rather than a single direction. It is designed for features that live on low-dimensional manifolds, which are especially frequent in vision. This work studies the BSF's strengths and weaknesses, finding how it still somewhat suffers from classic SAE failure modes, like feature splitting and composition. We propose several architectural changes to the BSF, including a Tournament Top-K selection rule that significantly reduces feature splitting, and we also extend the block paradigm to the crosscoder.

Figures & tables

Figure 1: Figure from Bhalla et al. (2026) . Here, the concept manifold is a helix. The first and third pictures show how SAEs capture manifolds. The second picture depicts the ideal capturing regime.

Figure 2: The Manifold Zoo: 128 features (64 segments + 64 curved manifolds), each embedded in $\mathbb{R}^{128}$ by a random orthonormal frame; each sample sums $|S|=4$ features.

Method	Indep.	Corr.
SAE (raw atoms)	0.36	0.36
SAE + post-hoc (ValuePCA)	0.60	0.41
SAE + post-hoc (partial-corr.)	0.65	0.59
BSF	0.77	0.77

Table 1: Mean recovery

R^{2}

of the BSF against a parameter-matched SAE, with and without post-hoc manifold recovery, on the independent and correlated datasets.

Figure 3: Per-seed recovery $R^{2}$ of each feature (columns) across 10 seeds (rows): segments, spheres, helices, and tori are almost always well recovered, while the other manifolds flicker from seed to seed.

Figure 4: Number of blocks each feature needs to reach $R^{2}>0.8$ , per seed: either 1, 2, or never (within 8).

Figure 5: Firing correlation between the blocks of two seeds: close to an identity, fading for blocks whose features tend to split.

Figure 6: The 10-seed experiment repeated on the correlated dataset: similar $R^{2}$ scores and similar feature splitting patterns as on the independent dataset.

Figure 7: Sweeping the number of blocks $G\in\{128,256,512\}$ : splitting persists in every setting, even at $G=M=128$ .

Figure 8: Sweeping sparsity $k\in\{4,6,8\}$ : performance degrades as $k$ increases; at $k=8$ every feature splits across multiple blocks.

Figure 9: Visualizing split features (one swiss roll, one helix): ground truth, what each of the two blocks captures alone, and their sum. The two blocks tend to be near-orthogonal complements.

Figure 10: Tracking 83 split block pairs through training: subspace overlap $\lVert D_{A}D_{B}^{\top}\rVert_{F}^{2}$ (left) and each block’s firing correlation with the feature (right).

Figure 11: Subspace overlap $\Omega$ through training for block pairs that are similar for a while but settle on different features (orange), pairs that actually split one feature (blue), and random pairs (gray), with the duel-opening ( $\Omega=0.1$ ) and permanent-verdict ( $\Omega=0.5$ ) thresholds.

Figure 12: Plain top-K vs. Tournament Top-K on the correlated dataset: recovery $R^{2}$ per feature and seed. Tournament Top-K lifts the fraction of well-recovered manifolds from $0.77$ to $0.90$ .

	DINO BSF	CLIP BSF
Blocks recovered	82%	53%
Pool (sh. / DINO / CLIP)	26 / 73 / 1	62 / 5 / 33

Table 2: Matching each single-model BSF’s blocks to the block-DFC crosscoder’s blocks by firing rate correlation: fraction of blocks recovered, and which crosscoder pool (shared / DINO-only / CLIP-only, in %) the recovered blocks come from.

Sparse Autoencoders

Sparse autoencoders decompose neural network activations into sparse features to study, interpret, and control learned representations.

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What is the geometry of a visual percept? The most widely used protocols for decomposing neural network representations into interpretable parts treat concepts as isolated directions, yet recent work shows that concepts are often realized as geometric structures in low dimensional regions of activation space. We turn to the literature of Structured sparsity to close this gap, and show that block sparsity, which groups directions into blocks, is the prior matched to a generative model in which a representation is a sparse sum of low-dimensional manifolds: the modern, learned form of a classical idea in visual neuroscience, where a visual feature is carried by a coordinated group of neurons rather than a single tuned one. We implement three variants of block-sparse featurizers (BSFs) and, through a minimum-description-length analysis, show that all three describe activations more compactly than direction-based featurizers, with the recovered concepts typically two- to four-dimensional. We then use BSFs to (i) recontextualize prior work, showing that curve detectors in InceptionV1 actually read from a single continuous curve manifold, (ii) discover novel manifolds including shadows and lighting in DINOv3, and (iii) support interpretable control of image generation in diffusion models (SDXL) via manifold steering.

Abstract

Sparse Autoencoders

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