Corrected Block Sparse Spectral Optimization: Exact Models, Finite Identification, and Local Dynamics

Omid Shams Solari

Figure 1: Printed-vs-corrected model map. The left column lists four defects of the seed-paper block formulation; the right column states the FACET correction that each defect motivates. The downstream theorems A1–A10 depend on all four corrections being in force simultaneously.

Figure 2: The corrected model family. A-SUM and A-SQ have different exact primal interpretations and are not silently identified.

Estimator	Primal (per column $j$ )	Reduced objective	Loading recovery / inactive policy
A-SUM	$\displaystyle\max_{\left\lVert z_{j}\right\rVert_{2}\leq 1}\!\!\bigl\{c_{j}^{\!\top}z_{j}-\gamma_{j}\left\lVert z_{j}\right\rVert_{1}\bigr\}$	$F_{\mathrm{sum}}(X)=\sum_{j}\left\lVert\mathcal{S}_{\gamma_{j}}(c_{j})\right\rVert_{2}$	$z_{j}^{\star}=\mathcal{S}_{\gamma_{j}}(c_{j})/\left\lVert\mathcal{S}_{\gamma_{j}}(c_{j})\right\rVert_{2}$ when nonzero; otherwise $z_{j}=0$ is the chosen inactive branch (loading-ball slack)
A-SQ	$\displaystyle\max_{z_{j}}\!\!\bigl\{2c_{j}^{\!\top}z_{j}-\left\lVert z_{j}\right\rVert_{2}^{2}-2\gamma_{j}\left\lVert z_{j}\right\rVert_{1}\bigr\}$	$F_{\mathrm{sq}}(X)=\sum_{j}\left\lVert\mathcal{S}_{\gamma_{j}}(c_{j})\right\rVert_{2}^{2}$	$z_{j}^{\star}=\mathcal{S}_{\gamma_{j}}(c_{j})$ (unconstrained quadratic lift; the vector itself is the loading, no renormalization)
A-HT (hard-threshold, exact cardinality $k$ )	$\displaystyle\max_{\left\lVert z_{j}\right\rVert_{2}\leq 1,\,\left\lVert z_{j}\right\rVert_{0}\leq k}\!\!c_{j}^{\!\top}z_{j}$	$F_{k}(X)=\sum_{j}\sum_{i\in T_{k}(c_{j})}c_{j,i}^{2}$	$z_{j}^{\star}$ supported on the top- $k$ coordinates of $c_{j}$ (deterministic tie policy); refit on the chosen support

Table 1: Corrected block estimator family and loading recovery. The three columns define distinct estimators; §3 proves A-SUM

\neq

A-SQ at

m\!\geq\!2

and A-HT is a third exact-cardinality variant. Loading recovery is the loading-column argmax at a fixed score and defines the raw estimand that the restricted refit stage then post-processes.

Edge case	Policy	Pinning test
Threshold tie ( $\|c_{i}\|=\gamma$ )	Treated as inactive ( $\mathcal{S}_{\gamma}(c)_{i}=0$ ) via the standard $(\cdot)_{+}$ convention; signed subgradient reused by the solver.	tests/test_paper_a_identities.py::test_a1_*
Zero soft-threshold vector ( $\mathcal{S}_{\gamma}(c)=0$ )	Loading set to zero; the loading-ball slack delivers primal value $0$ ; no renormalization attempted.	test_a1_sum_norm_zero_support_returns_zero
Mask-first refit on a non-empty support	$z=Mc/\left\lVert Mc\right\rVert_{2}$ ; empty support raises ZeroDirectionError rather than returning NaN.	tests/test_paper_a_edge_cases.py::test_a4_mask_then_normalize_*
Rank-deficient polar input (Stiefel oracle)	canonical policy completes via SVD; raise policy rejects and surfaces the assumption violation. Both treated as explicit theorem conditioning.	test_a5_canonical_policy_* , test_a5_raise_policy_*
Zero direction to a linear oracle	Both polar policies raise ZeroDirectionError unconditionally.	test_a5_zero_direction_raises_regardless_of_policy
Top- $k$ tie at the boundary (A-HT)	Lower original index wins; set-valued tie metadata recorded on TopKSelection .	test_topk_tie_policy_prefers_lower_index

Table 2: Edge-case policies in the FACET reference implementation. These are the operational conventions on which every proved theorem is explicitly conditional; running the solver under a different policy voids the theorem’s claim.

Assumption	Enables	Failure manifestation
Strict threshold margin	Finite support/sign identification	Persistent switching or tie dependence
Full-rank local polar argument	Smooth polar branch	Set-valued update and unstable derivative
Active eigengap or $\rho(D\Phi)<1$	Local linear convergence	Slow/noncontractive local dynamics
Certified distance radius	Safe screening	Heuristic false elimination if radius is guessed
Independent document units	Valid uncertainty summaries	Token-level pseudo-replication

Table 3: Assumption and failure-mode taxonomy.

Figure 3: Two-phase interpretation: discrete active-structure discovery followed by conditional smooth dynamics.

Figure 4: Safeguarded Riemannian ascent, the analyzed algorithm. Gradient

\to

tangent projection

\to

retraction

\to

Armijo-safeguarded acceptance with geometric step shrinkage. The stopping rule combines a tangent-residual criterion with both step-norm and support-stability tests; objective change alone is insufficient. Historical full-polar iteration omits the Armijo loop and carries no last-iterate claim.

Figure 5: The theorem dependency graph. Identification and local rates are not available before the model, algorithm, and convergence object are fixed.

Figure 6: Theory-to-evidence chain for the developed manuscript.

Figure 7: Validation stack. Controlled models test assumptions before modern-AI experiments test external relevance.

Figure 8: Q1, A-SUM and A-SQ disagree at

m\!\geq\!2

. Deterministic two-component instance

(A,\boldsymbol{\gamma})

from the A3 counterexample, with

X(\theta)

sweeping the

2{\times}2

rotation group. Left: the two reduced objectives have distinct arg-maximisers

\theta_{\mathrm{sum}}{\approx}30^{\circ}

and

\theta_{\mathrm{sq}}{\approx}72.5^{\circ}

. Right: each objective attains its own maximum at its own frame and strictly loses at the other, the ordering reversal that establishes Proposition 3.3 . The counterexample is deterministic rather than sampled; the fixture is pinned in tests/test_paper_a_identities.py::test_a3_sum_vs_squared_aggregation_rank_two_candidates_differently .

Figure 9: Q3, identification phase transition over (noise, margin fraction). Five replicates per cell on a

p{=}10

,

n{=}25

,

k{=}5

planted-spike instance with

\sigma_{1}{=}5

; safeguarded Riemannian ascent on A-SQ. Left: exact support-recovery rate across the (noise std, margin fraction) grid. Recovery is uniform at

\sigma_{\mathrm{noise}}{\leq}0.05

, degrades to

{\sim}0.6

at

0.15

, and collapses by

0.30

, consistent with the heuristic break-point

\sigma_{\mathrm{noise}}{\sim}c_{\min}/\sqrt{2\log(n-k)}

at which the strongest spurious off-support correlation exceeds the strict margin. Right: median iterations to support stabilisation among recovered runs; the solver cost grows modestly with noise until recovery fails. The vertical axis is the audit fix for a prior degenerate panel in which the clean rank-one setting trivialised every cell (plan 2026-10-13T1908Z §Q3).

Figure 10: Q2, safeguarded ascent trajectories on A-SQ. Per-iteration traces across five replicates on a

p{=}10

,

n{=}25

,

k{=}5

planted-spike instance with margin fraction

0.6

and initial angle

0.6

rad; one trace per colour. Left: monotone increase of

F_{\mathrm{sq}}(X_{k})

(empirical witness of Lemma 6.5 , A6). Centre: step norm

\left\lVert X_{k+1}-X_{k}\right\rVert_{F}

on a log axis; the geometric decay to a numerical floor exhibits the bounded total variation that Theorem 6.7 (A8) requires for whole-sequence Cauchy convergence. Right: tangent-residual

\|\mathrm{grad}_{R}F(X_{k})\|

on a log axis, the computable stationarity measure that pins Lemma 6.6 (A7). Formal proofs are in Section 6 ; one seed drops out (Armijo stall near the fixed point at tolerance

10^{-10}

).

Figure 11: Q4, local-rate prediction versus observation: two branches. Hand-constructed rank-two operator with known top-two restricted eigenvalues

(\lambda_{1},\lambda_{2})

on the identified support;

\lambda_{1}

fixed and

\lambda_{2}

swept. The observed geometric rate is read from a log-linear fit of

\sin\angle(x_{t},x_{\star})

over the strictly decaying prefix of the trajectory. (a) Pure power-iteration branch (

\gamma{=}0

): the sphere linear-oracle iteration coincides with power iteration on

A_{S}A_{S}^{\top}

and the observed rate matches the classical prediction

\lambda_{2}/\lambda_{1}

to within

{\sim}5\%

across the sweep. (b) A-SQ branch with the standard shrinkage

\gamma{=}\gamma^{\star}

: the gradient

A_{S}(A_{S}^{\top}x-\gamma\,\mathrm{sign}(c))

is a shifted power iteration whose effective denominator in the invariant two-dimensional subspace is

\lambda_{1}-\gamma\,\sigma_{1}\sqrt{k}

, giving the corrected prediction

\lambda_{2}/(\lambda_{1}-\gamma\,\sigma_{1}\sqrt{k})

. Observed rate matches to within

1\%

except at the near-divergence boundary where the Armijo safeguard truncates. Both panels share the

y{=}x

reference. Replaces an earlier single-branch panel whose dashed reference was systematically off by the shrinkage factor (plan 2026-10-13T1908Z §Q4).

Figure 12: A4-refit, restricted refit dominates raw selection. Scatter of refitted vs raw restricted objectives on the identified support, across clean rank-one planted spikes (noiseless and noisy) and controlled rank-two cells parameterised by

\lambda_{2}/\lambda_{1}

. Left: every point lies at or above the dashed

y{=}x

line, the A4 dominance invariant. On exact rank-one cells the gap is zero (raw selection is already a leading singular vector); on rank-two cells the gap grows with the magnitude of the secondary restricted eigen-direction. Right: the gap

(\mathrm{refit}-\mathrm{raw})

box-plotted against the noise axis; the box widens with noise as rank-one planted supports pick up a nontrivial second component. Dominance pinned in tests/papers/a/test_a4_smoke.py .

noise $\sigma$	margin frac	n	support rec.	raw $\\|A_{S}z_{\mathrm{raw}}\\|_{2}$	refit $\sigma_{1}(A_{S})$	refit $-$ raw
-0.30	0.75	10	0.00	1.813	1.816	0.003
-0.20	0.50	10	0.10	1.866	1.868	0.002
-0.10	0.25	10	1.00	2.000	2.000	0.000
0.00	0.30	10	1.00	2.000	2.000	0.000
0.00	0.60	10	1.00	2.000	2.000	0.000
0.10	0.30	6	0.83	2.030	2.030	0.000

Table 4: Raw selection vs restricted refit on the identified support. Each row aggregates n replicates per (noise, margin) cell on the the BGSP paper planted-spike generator; mean across replicates.

Figure 13: T2, controlled-neural bridge: feature recovery across three sparsity regimes. Elhage-style compressed autoencoder with

n{=}12

planted features squeezed into

d{=}6

hidden dimensions, trained to minimise an importance-weighted reconstruction loss ( Elhage et al., 2022 ) . Sparsity is swept across three regimes (very sparse

\approx 1

feature active per sample, moderate

\approx 4

, dense

\approx 8

). We report the Hungarian bipartite-matched

|\cos(w_{i},x_{\sigma(i)})|

: each component is assigned to at most one planted feature, so a single component cannot cover multiple features; the lenient

\max_{j}|\cos|

metric previously reported in this slot is inflated by antipodal collapse. The on-figure annotation “

\mathrm{unique}{=}u

” is the number of distinct planted directions after collapsing antipodal pairs. At low density the Elhage toy learns pair-antipodal arrangements (

u{=}d{=}6

) and all three methods are bounded by the same unique-direction budget, with FACET recovering every unique direction exactly while PCA mixes them under importance decay. In the dense regime all

u{=}12

planted directions are distinct, which no

m{=}d

orthonormal basis can match; the ceiling that remains is the expected Stiefel limitation and motivates the over-complete comparator of Figure 19 .

Figure 14: T3, GPT-2 small residual-stream activations. GPT-2 small (

d{=}768

, 12 transformer blocks) is run on ten public-domain neutral paragraphs (

N{\approx}920

tokens total); post-block-6 residual activations are decomposed with

m{=}32

components by FACET A-SQ (safeguarded Riemannian ascent,

\gamma{=}0.15

), PCA (top-

m

eigenvectors of

HH^{\top}

), and a random orthogonal baseline. Top row: reconstruction

R^{2}

, mean Hoyer sparsity per component, mean score kurtosis per component. FACET and PCA both recover

\approx\!98\%

of the Frobenius energy at matched

m

and produce near-identical score-distribution statistics; the random baseline stops near

m/d\approx 0.04

and inherits a highly non-Gaussian score distribution from the heavy-tailed residual-stream activation norms. Bottom row: multi-probe causal-ablation battery; for the top-eight components of each method we ablate one component at a time from the residual stream and record

\mathrm{KL}(p_{\mathrm{clean}}\|p_{\mathrm{abl}})

at the final token position across ten held-out probe sentences (

8\!\times\!10{=}80

measurements per method). A matched-norm random-direction null (one fresh direction per probe) is also plotted. FACET and PCA components exhibit comparable median effect but a long tail reaching 4–5 bits of KL, roughly two orders of magnitude above the random-direction null; the previous single-probe single-component panel underestimated this spread (plan 2026-10-13T1908Z §T3).

Figure 15: M1, FACET

-

PCA gap across GPT-2 small layers and hook sites. For each block

L\in\{0,2,4,6,8,10,11\}

and hook site

s\in\{\texttt{resid\_post},\texttt{attn\_out},\texttt{mlp\_out}\}

we extract activations on the T3 corpus and fit

m{=}32

components with FACET A-SQ (safeguarded) and PCA; heat maps show (FACET

-

PCA) for three metrics with red indicating FACET larger. Left: reconstruction

R^{2}

is within

[-0.01,0]

of PCA on every cell, FACET pays no representational cost. Centre: FACET is uniformly sparser in Hoyer, with the largest gain at attn_out in blocks

L\in[2,6]

and at mlp_out of block

0

. Right: score-kurtosis gap is positive on attn_out / mlp_out (FACET concentrates activity into fewer samples) and modestly negative on resid_post (FACET smooths across the full residual stream). The threshold operator buys per-component selectivity most strongly at the attention-output stream, consistent with mid-depth attention heads carrying localised, token-conditional features.

Figure 16: M2, top-token coherence (GPT-2 block 4 attn_out ). For each of the

m{=}32

components in each method we take the top-

K{=}10

tokens by projected score

|x_{j}^{\top}h_{t}|

and measure the mean pairwise cosine of their GPT-2 input ( wte ) embeddings. Higher coherence = more semantically coherent exemplars under the input-embedding geometry = more monosemantic component by that proxy (SAEBench-style evaluation ( Karvonen et al., 2025 ) ). Left: per-component distribution as a box plot with individual components over-plotted. Right: histograms of the same distribution. FACET components have higher median coherence (0.28 vs PCA 0.26) and a longer upper tail (max 0.51 vs 0.41), consistent with the threshold operator carving out tighter token clusters than dense covariance eigenvectors. The top tokens of the top-six components of each method are emitted alongside the figure as m2_exemplars.json .

Figure 17: M6, classical sparse-PCA benchmark on Jeffers Pitprops ( Jeffers, 1967 ) ; thirteen-variable correlation data,

m{=}6

components. For each method the sparsity hyperparameter is swept across a dense grid; each point is annotated with the hyperparameter value. Horizontal axis: total nonzeros across the six loadings (dense fill

={}13\times 6{=}78

). Vertical axis: adjusted explained variance in the Shen–Huang sense ( Shen & Huang, 2008 ) ,

\operatorname{tr}(Q^{\top}CQ)

where

Q

is the orthonormal basis of the sparse-loading subspace; this correctly handles non-orthogonal loadings whereas the naive sum of per-component Rayleigh quotients double-counts overlapping support. FACET A-SQ and A-SUM lie on, or just below, sklearn ’s SparsePCA (the elastic-net analogue of Zou et al. ( Zou et al., 2006 ) ) across the whole Pareto frontier and recover the dense PCA ceiling (

{\approx}\,10.986

) by

\mathrm{nnz}\!\approx\!60

. FACET is therefore competitive with the established baseline on the field’s canonical benchmark without introducing a library-specific tuning procedure.

Figure 18: M7, A10 safe-screening payoff: advisory ceiling vs measured solver savings. Planted spikes with

p{=}200

,

n{=}2000

,

k\in\{100,200,400,800,1200,1600\}

, five replicates per level. Top row (advisory): the safeguarded ascent runs with the standard DenseOperator and we apply Proposition 5.4 at each iterate using the ground-truth distance

\|X_{k}-X_{\star}\|_{2}

as the certified radius. Per-iteration curves (left) climb monotonically to the

(n{-}k)/n

ceiling as the certified ball shrinks; final-iteration values (right) hit the ceiling exactly at every sparsity. Bottom row (measured solver uptake): the same mask is plugged into the solver via a dedicated ScreenedDenseOperator that drops

A^{\top}X

cost from

\mathcal{O}(np)

to

\mathcal{O}(n_{\mathrm{active}}\,p)

. Measured oracle-work reduction (red) tracks the ceiling (dashed black) across the sparsity grid; wall-clock speedup is

1.40\times

at

k/n{=}0.05

and

1.36\times

at

k/n\in\{0.1,0.2\}

. Objective values of full and screened solves agree to machine epsilon (

<10^{-14}

), confirming the safety claim as a solver-level invariant.

Figure 19: M4, FACET vs tied sparse autoencoder at matched

m{=}32

on GPT-2 small block 4 attn_out . A tied-decoder SAE (dictionary size

={}m

,

L_{1}

coefficient

0.01

, 3000 Adam steps on the T3 corpus activations) is compared head-to-head with FACET A-SQ and PCA on four metrics evaluated at identical

m

. Reconstruction

R^{2}

: PCA

0.574

, FACET

0.573

(tied), SAE

0.529

, the SAE’s

L_{1}

penalty costs

{\sim}\,8

percentage points of representation. Mean Hoyer sparsity per component: SAE

0.324

, FACET

0.293

, PCA

0.234

. Median monosemanticity (M2 coherence): FACET

0.278

, SAE

0.269

, PCA

0.263

. Mean causal-ablation KL across five probes

\times

top-six components: PCA

1.40

bits, FACET

1.27

, SAE

1.27

. Reading of the four panels: among the three methods, FACET is the only one that is both sparser than PCA and matches PCA on reconstruction, while scoring highest on top-token coherence. The SAE pays a reconstruction cost for its

L_{1}

-driven sparsity.

Figure 20: M5, ViT CLS-token decomposition on ten synthetic image classes. Ten rendered class templates (solid colours, horizontal/vertical stripes, diagonal gradient, centred disks, frames; twenty jittered variants each, so

200

images) are passed through pretrained ViT-base-patch16-224 and the final-block CLS tokens form

H\in\mathbb{R}^{768\times 200}

. FACET A-SQ and PCA fit

m{=}10

components. Left: reconstruction

R^{2}\approx 0.968

for both methods (tied). Middle: adjusted mutual information between the dominant-component assignment (

\arg\max_{j}|x_{j}^{\top}h_{i}|

) and the ground-truth class label. FACET:

0.64

(strong agreement). PCA:

{\approx}\,0

, the dominant PCA component is the same high-variance direction for almost every image, so the clustering collapses. Right: silhouette score of the raw CLS tokens, computed in the full

768

-dim embedding space; comparable for both (

\approx 0.86

). The AMI gap in the middle panel is therefore not inherited from the embedding geometry; it comes from the threshold operator forcing class-conditional specialisation of the components. This modality-agnostic result directly witnesses the identification theorem (Theorem 5.2 ) on image data.

Figure 21: M8: sparse risk factors on Fama-French 48 industry daily returns. 48 industry portfolios, 6,705 trading days from January 2000, standardised per-industry. (a) Top-four FACET A-SQ components at

\gamma{=}0.40

, drawn as per-industry loading bars sorted by

|

weight

|

and coloured by the GICS-style broad sector; the “

-

” badge marks a negative loading. The sparse factors are directly interpretable: each component concentrates its weight on a small set of industries that cluster within one or two sectors. (b) Pareto frontier of (total nonzeros across the

m{=}6

loadings, Shen–Huang adjusted explained variance). FACET A-SQ and sklearn SparsePCA (the Zou-style elastic-net analogue) track dense PCA’s ceiling to within

1\%

of adjusted EV while cutting

20{-}50\%

of the loading entries. Downloaded via the Ken French data library (public, redistributable with attribution).

Figure 22: M9: single-cell RNA-seq on 10x Genomics PBMC 3k, matched-sparsity sweep. 2,700 PBMCs

\times

2,000 highly-variable genes after per-cell total-count normalisation and

\log 1p

; ground-truth cluster labels come from PCA +

k

-means on 20 PCs, named by the canonical PBMC marker-gene programme (T CD4+/CD8+, NK, B, mono CD14+/FCGR3A+, DC, platelet). (a) Cluster-recovery AMI (dominant-component

\to

class) vs total nonzeros across all

m{=}8

loadings, four methods swept over sparsity. Dense PCA (blue square) sets the ceiling at AMI

0.606

; FACET A-SQ (red) and sklearn SparsePCA (green) sit near this ceiling at near-dense sparsity, and truncated PCA (cyan) the ad-hoc baseline of hard-thresholding dense loadings, degrades steeply below

\mathrm{nnz}{\lesssim}600

. (b) Silhouette in the top-two variance-dominant component plane. FACET A-SQ attains the highest silhouette of any method (

0.291

at

\gamma{=}0.03

) exceeding dense PCA (

0.208

) SparsePCA’s best (

0.276

) and every truncated-PCA cell. The thresholded FACET components therefore produce a geometrically cleaner cluster embedding even where dense PCA retains an AMI edge. (c) Top-15 gene loadings per component for FACET at the hero

\gamma

. Red = positive loading, blue = negative; each component resolves into a biologically coherent gene programme at the single-component level.

Figure 23: M9 interpretability hero: PBMC 3k with pseudo-labels. Same 2,700 PBMCs

\times

2,000 HVGs as Fig. 22 ; the ground-truth labels are the PCA +

k

-means + marker-program pseudo-labels used throughout M9. The FACET gamma shown is the interpretability selection (sparsest FACET with AMI within

0.025

of the max-AMI FACET), not the max-AMI gamma. (a) Mean marker-gene enrichment of the top-

K

loadings per component, averaged across components, with per-component inter-quartile band and the uniform-random baseline. FACET and PCA track each other closely because the pseudo-labels are PCA-derived; the win is in gene economy, FACET uses only

551

active genes per component against PCA’s

2000

(

3.6\times

denser). (b) FACET cell-type purity heatmap (mean

|

score

|

,

z

-scored per component). Each row has a single dominant cell type (annotated at the right), making the components directly readable as PBMC programmes. (c) PCA cell-type purity on the same colour scale is markedly diffuse. The pseudo-label caveat is relaxed on native labels in Figs. 25 , 27 , and 29 .

Figure 24: M9 Zheng17 FACS-sorted PBMCs, native cell-type labels. Eight bead-purified subpopulations (CD14+ Monocyte, CD34+ HSPC, CD4+ Helper T, CD56+ NK, Memory T, Naive Cytotoxic, Naive T, Regulatory T) concatenated at

2{,}000

cells per population, preprocessed with the same

\log 1p

+ HVG +

z

-score pipeline as M9. Labels are the FACS gates, independent of any clustering step. (a) AMI vs total nonzeros across all

m{=}8

loadings. FACET A-SQ attains AMI

0.401

, above the dense-PCA ceiling at

0.378

, flipping the ranking seen on PCA-derived pseudo-labels in Fig. 22 . (b) Silhouette in the top-two component plane. (c) Top-15 gene loadings per component at the hero FACET gamma.

Figure 25: M9 Zheng17 FACS interpretability hero. Companion to Fig. 24 , interpretability-first panels on the same bead-purified data. (a) Mean enrichment vs top-

K

in the Zheng FACS marker list (CD14+ Mono, CD34+, CD4 helper, CD56+ NK, Treg, memory/naive T subsets); gene-economy banner reports the per-component active-gene budget. (b) FACET cell-type purity heatmap with dominant FACS gate annotated per component (right-hand labels). Each FACET component localises on a single FACS population. (c) PCA cell-type purity on the same colour scale is visibly more diffuse.

Figure 26: M9 Zheng17 68k PBMC (donor A), native cell-type labels. 68,551 cells

\times

2,000 HVGs; the per-barcode labels are the FACS-correlated cell-type assignments distributed with the Zheng et al. 2017 analysis repository, not derived from any clustering of this matrix.

m{=}10

components. (a) AMI vs total nonzeros: FACET A-SQ reaches AMI

0.243

against the dense-PCA ceiling of

0.219

(

+0.024

). (b) Silhouette in the top-two component plane. (c) Top-15 gene loadings per component at the hero FACET gamma; each component resolves into an immune programme consistent with its dominant cell type.

Figure 27: M9 Zheng17 68k interpretability hero. Companion to Fig. 26 . (a) Mean enrichment vs top-

K

in the PBMC marker list; the gene-economy annotation reports FACET’s active-gene budget per component against PCA’s. (b) FACET per-component cell-type purity heatmap with the dominant FACS-correlated cell type labelled on each row. Components

c1

through

c10

resolve into Dendritic / CD14+ Mono / CD56+ NK / CD19+ B / CD4+ Memory / Treg / CD4+ Naive and related programmes. (c) PCA purity on the same colour scale is diffuse and does not admit a clean cell-type reading per component.

Figure 28: M9 Tabula Muris FACS (Marrow + Spleen + Thymus), native mouse cell-type labels. 8,083 cells

\times

2,000 HVGs across

25

cell_ontology_class categories; labels come from the official Tabula Muris annotations, not from any clustering of this matrix.

m{=}10

components. (a) AMI vs total nonzeros: FACET A-SQ reaches AMI

0.617

against the dense-PCA ceiling of

0.556

(

+0.061

, the largest margin of the three native-label scRNA experiments). (b) Silhouette in the top-two component plane. (c) Top-15 gene loadings per component at the hero FACET gamma; mouse symbols (title-case).

Figure 29: M9 Tabula Muris interpretability hero. Companion to Fig. 28 , interpretability-first panels on the same mouse Smart-seq2 data. (a) Mean enrichment vs top-

K

in the mouse immune / stromal marker list; gene-economy annotation reports per-method active-gene budget per component. (b) FACET cell-type purity heatmap with dominant cell_ontology_class annotated per row. (c) PCA cell-type purity on the same colour scale.

Figure 30: M10: agentic AI, component–phase heatmap on multi-turn tool-use conversations. Seven hand-written conversations alternate through greeting , planning , tool_call , tool_response , reasoning , answer , and summary phases (

836

tokens total). GPT-2 small extracts residual-stream activations per token at block 6; PCA (a) and FACET A-SQ (b) fit

m{=}16

components. Each cell is the mean

|

component score

|

for tokens in that phase, standardised per component (row

z

-score). Rows are sorted by peak phase magnitude. The in-panel badges report the dominant-component AMI against the ground-truth phase label and the mean per-component “phase selectivity”

(\max-\mathrm{median})/\max

. FACET components concentrate activity on tool_call , tool_response , and greeting spikes (red/blue bands) while PCA spreads activity more uniformly across phases.

Corrected Block Sparse Spectral Optimization: Exact Models, Finite Identification, and Local Dynamics

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