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\title[Dimensional Collapse / ASI]{Dimensional Collapse in Transformer Attention Outputs}
\subtitle{A Challenge for Sparse Dictionary Learning}
\author{Paper: Junxuan Wang, Xuyang Ge, Wentao Shu,\\ Zhengfu He, and Xipeng Qiu}
\date{Reading seminar \enspace|\enspace arXiv:2508.16929v4 \enspace|\enspace 11 November 2026}
\newcommand{\fig}[2]{\centering\includegraphics[width=\linewidth,height=#2,keepaspectratio]{figures/#1.pdf}\par}
\begin{document}
\begin{frame}
\titlepage
\end{frame}
\begin{frame}{Data geometry can make SAE capacity go unused}

\begin{block}{Central idea}
Initialize sparse features in the data's active subspace, then train normally.
\end{block}
\begin{itemize}
\item Attention outputs often have a concentrated singular spectrum.
\item ASI improves reconstruction and reduces dead features.
\item At 1M features, the $87\%\to{<}1\%$ result uses \alert{ASI + SparseAdam}.
\end{itemize}
\medskip
\small Published at \href{https://proceedings.mlr.press/v306/wang26bh.html}{ICML 2026, PMLR 306}.\quad
\href{https://github.com/OpenMOSS/Llamascopium}{Official code: Llamascopium}.\\
Read version: \href{https://arxiv.org/abs/2508.16929v4}{arXiv v4, 11 February 2026}; links checked 11 November 2026.

\papersource{Paper, Abstract and Secs. 4--5, PDF pp. 1, 4--8.}
\end{frame}
\begin{frame}{Why should random features fill the entire space?}

\begin{columns}[c]
\begin{column}{0.57\textwidth}
\fig{head_pic_icml}{4.5cm}
\end{column}
\begin{column}{0.4\textwidth}
\begin{itemize}
\item Larger dictionaries can waste memory and training on inactive features.
\item A \alert{dead feature} never activates over 10M evaluation tokens.
\item The proposed intervention changes the starting geometry.
\end{itemize}
\end{column}
\end{columns}

\papersource{Paper, Fig. 1 and Sec. 2.3, PDF pp. 2--3.}
\end{frame}
\begin{frame}{The target is the output written to the residual stream}

\[
Z_i=\operatorname{softmax}\!\left(\frac{Q_iK_i^\top}{\sqrt{d_h}}\right)V_i,
\quad Z=[Z_1,\ldots,Z_H]
\]
\[
\boxed{O=ZW^O=\sum_{i=1}^H Z_iW_i^O}
\]
\begin{itemize}
\item $Z_i\in\mathbb R^{n\times d_h}$; $Z,O\in\mathbb R^{n\times d}$; $W^O\in\mathbb R^{d\times d}$.
\item $O$ is the attention block's total write into the residual stream.
\item Keep attention patterns, concatenated heads $Z$, and outputs $O$ distinct.
\end{itemize}

\papersource{Paper, Sec. 3.1, PDF p. 3. Standard MHSA notation: $Hd_h=d$.}
\end{frame}
\begin{frame}{Effective rank summarizes how concentrated a spectrum is}

\[
\widetilde A=U\Sigma V^\top,\qquad
p_j=\frac{\sigma_j}{\sum_k\sigma_k},\qquad
\operatorname{erank}(\widetilde A)=\exp\!\left(-\sum_j p_j\log p_j\right)
\]
\begin{itemize}
\item $\widetilde A$: mean-centered token activations, shape $n\times d$.
\item Uses \alert{singular values}, rather than squared singular values.
\item Equal singular values give the full rank; concentrated spectra lower it.
\item $\operatorname{erank}/d\approx0.6$ describes spectral concentration; it does not imply that the remaining directions are exactly zero.
\end{itemize}

\papersource{Paper, Definition 4.1, Sec. 4.1, PDF p. 4.}
\end{frame}
\begin{frame}{Attention outputs typically use fewer effective directions}

\fig{effective_rank}{3.8cm}
\begin{itemize}
\item Left: layers. Middle: models. Right: data domains.
\item Main-text summary: attention $\sim60\%$, MLP/residual $\sim90\%$ effective rank.
\item These are empirical trends; ratios vary by model, data, and layer.
\end{itemize}

\papersource{Paper, Fig. 2, PDF p. 4; counterexamples to a strict ordering appear in Fig. 10, PDF p. 15.}
\end{frame}
\begin{frame}{Task-relevant attention contributions are also concentrated}

\begin{columns}[c]
\begin{column}{0.47\textwidth}\fig{rsv}{3.4cm}\end{column}
\begin{column}{0.47\textwidth}\fig{dim_ablate_downstream_loss}{3.4cm}\end{column}
\end{columns}
\[
\mathrm{FLR}(m)=\frac{L_{\rm zero}-L_{\rm recovered}(m)}{L_{\rm zero}-L_{\rm original}}
\]
\small For $>99\%$ recovery: attention needs $39.1\%$ of dimensions, MLP $95.3\%$, residual $96.9\%$. Recovery is relative to the zero-ablation loss gap.

\papersource{Paper, Fig. 3, Sec. 4.3, PDF p. 5; App. B, PDF p. 14. Llama-3.1-8B, layer 15, SlimPajama.}
\end{frame}
\begin{frame}{The output projection further suppresses the spectral tail}

\begin{columns}[c]
\begin{column}{0.49\textwidth}\fig{z_wo}{4.5cm}\end{column}
\begin{column}{0.48\textwidth}
For a unit output direction $\hat e$, let
\[v=W^O\hat e,\qquad \hat v=v/\|v\|_2.\]
\[
\operatorname{Var}(O\hat e)
=\underbrace{\operatorname{Var}(Z\hat v)}_{Z}
\underbrace{\|v\|_2^2}_{W^O}.
\]
Tail suppression tracks $W^O$ strongly. Overlapping head write subspaces provide an intuitive explanation.
\end{column}
\end{columns}

\papersource{Paper, Sec. 4.4 and Fig. 4, PDF p. 5. Variance factors correspond to squared spectral scales; assume $v\ne0$.}
\end{frame}
\begin{frame}{Lower effective rank co-occurs with more dead features}

\fig{dead_erank_combined}{3.5cm}
\begin{itemize}
\item LlamaScope: same dictionary size 32,768 and $L_0=50$ across sites.
\item Attention outputs and late MLP outputs show the strongest co-occurrence.
\item Capacity and sparsity sweeps preserve the broad trend.
\item Sec. 6 leaves the precise causal dynamics unresolved.
\end{itemize}

\papersource{Paper, Fig. 5, PDF p. 6; App. F, Tables 1--3, PDF pp. 18--19; Sec. 6, PDF p. 8.}
\end{frame}
\begin{frame}{ASI folds a data-dependent projection into initial weights}

\[
V_{\rm active}=V_{:,:d_{\rm init}},\qquad
P=V_{\rm active}V_{\rm active}^{\top}
\]
\[
\boxed{W_D\leftarrow W_DP,\qquad W_E\leftarrow W_D^\top}
\]
\begin{enumerate}
\item Center representative activations and compute their SVD.
\item Select the leading $d_{\rm init}$ right singular directions.
\item Project the random dictionary; tie the encoder initialization.
\item Train the full SAE normally, without repeated projection.
\end{enumerate}
\small Row-vector convention: $W_D\in\mathbb R^{h\times d}$, $W_E\in\mathbb R^{d\times h}$; $h$ is dictionary size.

\papersource{Paper, Sec. 5.2, PDF p. 6; App. K, PDF p. 22. The displayed matrix update governs the whole dictionary.}
\end{frame}
\begin{frame}{Initialization aligns features without fixing a permanent bottleneck}

\[
z=\operatorname{TopK}(xW_E+b_E),\qquad \hat x=zW_D+b_D
\]
\[
\mathcal L=\|x-\hat x\|_2^2+\alpha\mathcal L_{\rm aux}
\]
\begin{itemize}
\item Ordinary TopK uses only the reconstruction term; AuxK adds an auxiliary loss.
\item ASI changes the starting directions; weights may subsequently leave the subspace.
\item $d_{\rm init}=d$ recovers ordinary initialization.
\item Main setting: $d=4096$, $d_{\rm init}=768$. The latter is an optimization choice, not the effective rank.
\end{itemize}

\papersource{Paper, Secs. 3.2, 5.2 and App. E, PDF pp. 3, 6, 16--18.}
\end{frame}
\begin{frame}{The main comparison holds capacity and training data fixed}

\begin{tabular}{ll}
\toprule
Target & Llama-3.1-8B, attention output, layer 15 (zero-based)\\
Data / training budget & SlimPajama / 800M tokens per SAE\\
Dictionary / sparsity & 32,768 features / $k=50$, unless swept\\
Batch / context & 4096 / 1024 tokens\\
Optimizer & Adam; SparseAdam added in scaling experiments\\
ASI / AuxK & $d_{\rm init}=768$ / $\alpha=1/32$, $k_{\rm aux}=512$\\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item Evaluate dead features, Normalized MSE, and $\Delta$ LM Loss; all lower is better.
\item Three seeds for sparsity curves; a separate 15-seed significance test.
\end{itemize}

\papersource{Paper, App. E--H, PDF pp. 16--20. Delta LM Loss: evaluated over 1M tokens.}
\end{frame}
\begin{frame}{At 32K features, ASI improves utilization and reconstruction}

\centering
\begin{tabular}{lrrr}
\toprule
Method & Dead count $\downarrow$ & NMSE $\downarrow$ & $\Delta$ LM Loss $\downarrow$\\
 & & & ($\times10^{-3}$)\\
\midrule
TopK & $16144.33$ & $0.33367$ & $8.375$\\
AuxK & $54.67$ & $0.32241$ & $7.958$\\
ASI & \alert{4.00} & \alert{0.31680} & \alert{7.847}\\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item $k=50$; means over three seeds. ASI dead-count std: $1.73$.
\item NMSE improves by $5.06\%$ relative to TopK, $1.74\%$ relative to AuxK.
\item The LM-loss increase falls by $0.000528$ vs. TopK; this is a much smaller absolute effect.
\end{itemize}

\papersource{Paper, Table 4, PDF p. 20; relative percentages calculated from reported means.}
\end{frame}
\begin{frame}{Moderately restricting the initial subspace works best}

\fig{d_init}{3.7cm}
\begin{itemize}
\item Horizontal axis: initial subspace dimension; all vertical metrics lower is better.
\item Full-space initialization leaves many features dead.
\item Very small subspaces damage reconstruction; 256--2048 is broadly usable.
\item Experiments use 768 by default, rather than a fixed fraction of effective rank.
\end{itemize}

\papersource{Paper, Fig. 6, PDF p. 6; App. J.3, PDF p. 22. Curves show three-seed mean and standard deviation.}
\end{frame}
\begin{frame}{The data-supported directions matter, not dimensionality alone}

\fig{d_init_svd_random}{3.8cm}
\begin{itemize}
\item Green: leading singular directions. Orange: random subspace.
\item Random restrictions worsen dead features, reconstruction, and LM loss.
\item Near-full-rank residual streams gain little from ASI.
\end{itemize}

\papersource{Paper, App. J, Fig. 12, PDF p. 21; residual-stream control: Fig. 13, PDF p. 22.}
\end{frame}
\begin{frame}{At large capacity, alive-feature counts do not tell the full story}

\fig{scaling}{3.8cm}
\begin{itemize}
\item Left: ASI improves reconstruction across 16K--1M total features.
\item Middle: ASI alone retains many dead features at 1M; AuxK keeps more alive.
\item Right: ASI uses alive features more efficiently than AuxK.
\end{itemize}

\papersource{Paper, Sec. 5.3 and Fig. 8, PDF pp. 7--8. Shared legend retained from the original figure.}
\end{frame}
\begin{frame}{SparseAdam complements geometry at the million-feature scale}

\begin{block}{The headline result has two ingredients}
Standard TopK: $\sim87\%$ dead at 1M features.\\
ASI + SparseAdam: $<1\%$ dead, with the lowest reconstruction error.
\end{block}
\begin{itemize}
\item ASI addresses the initial mismatch with data directions.
\item SparseAdam masks updates of parameters and moments to nonzero-gradient positions.
\item The paper links this benefit to stale-momentum effects in sparse training.
\item Optimizer learning rates were tuned separately: Adam $4\!\times\!10^{-5}$, SparseAdam $6\!\times\!10^{-5}$.
\end{itemize}

\papersource{Paper, Sec. 5.3, Fig. 8, App. E and M, PDF pp. 7--8, 16, 24--25.}
\end{frame}
\begin{frame}{Reconstruction gains transfer more consistently than minimum death}

\small
\centering
\begin{tabular}{lrrr}
\toprule
Target & Dead: AuxK / ASI & NMSE: AuxK & NMSE: ASI\\
\midrule
Llama / layer 7 & $98\;/\;5308$ & 0.29800 & \textbf{0.28882}\\
Llama / layer 23 & $332\;/\;27$ & 0.20235 & \textbf{0.20141}\\
Qwen / layer 8 & $4\;/\;7$ & 0.28849 & \textbf{0.28533}\\
Qwen / layer 26 & $66\;/\;566$ & 0.28127 & \textbf{0.28087}\\
\bottomrule
\end{tabular}
\normalsize
\medskip
\begin{itemize}
\item Llama-3.1-8B: SlimPajama; Qwen3-8B: fineweb-edu.
\item ASI has lower NMSE and LM-loss degradation in all four configurations.
\item AuxK has fewer dead features in three of the four configurations.
\end{itemize}

\papersource{Paper, App. I, Tables 6--7, PDF p. 21. All targets are attention outputs.}
\end{frame}
\begin{frame}{Lorsa benefits from head-aware subspace initialization}

\fig{l0_lorsa}{3.4cm}
\begin{itemize}
\item Group sparse Lorsa units by their original MHSA head.
\item Initialize input/output directions in the corresponding active subspaces; copy original Q/K parameters.
\item With 32,768 units, ASI improves reconstruction and dead-unit rate.
\item This evaluates the combined Lorsa initialization design.
\end{itemize}

\papersource{Paper, Sec. 5.4, Fig. 9, PDF p. 8; implementation details in App. K/O, PDF pp. 23, 26--27.}
\end{frame}
\begin{frame}{JumpReLU benefits when training starts with low activity}

\fig{mse+dead-l0-jumprelu-ijt2}{3.8cm}
\small Blue: base; green: ASI; red: the paper's Pre-Act Loss comparator.\par\normalsize
\begin{itemize}
\item Left: normalized reconstruction error. Right: dead-feature fraction.
\item ASI improves both metrics when $L_0$ starts low and stays low.
\item Larger $L_0$ means more active features per token.
\end{itemize}

\papersource{Paper, App. E.4/L, Fig. 14, PDF pp. 18, 24. JumpReLU, Llama-3.1-8B middle-layer attention output.}
\end{frame}
\begin{frame}{A dense training start can erase the initialization advantage}

\fig{mse+dead-l0-jumprelu-ijt1.1}{3.8cm}
\small Blue: base; green: ASI; red: the paper's Pre-Act Loss comparator.\par\normalsize
\begin{itemize}
\item High initial $L_0$, then reduced: ASI and base reconstruction nearly coincide.
\item High-$K$ annealing in TopK also removes the observed advantage.
\item Authors' explanation: early dense updates move features out of the subspace.
\end{itemize}

\papersource{Paper, App. L, Figs. 15--16, PDF pp. 24--25. High-$K$ TopK is a separate control.}
\end{frame}
\begin{frame}{Feature quality looks similar in a small blinded evaluation}

\begin{columns}[t]
\begin{column}{0.47\textwidth}
\textbf{Semantic consistency}\par\medskip
100 sampled features total; 50 per method. Scores range from 1 to 5.\\\medskip
\begin{tabular}{lrr}
\toprule
Method & Score 5 & Mean*\\
\midrule
ASI & 17/50 & 3.54\\
TopK & 18/50 & 3.52\\
\bottomrule
\end{tabular}
\end{column}
\begin{column}{0.48\textwidth}
\textbf{Low-activity directions}\par\medskip
Reconstruction error projected onto the paper's dead subspace:\\\medskip
\begin{tabular}{lr}
\toprule
TopK & 0.00350\\
ASI & 0.00334\\
\bottomrule
\end{tabular}\par
\medskip
Similar errors support retained coverage in this comparison.
\end{column}
\end{columns}
\medskip More alive features do not establish better monosemanticity.

\papersource{Paper, App. N, Tables 8--10, PDF pp. 25--26. *Means calculated from score counts; authors report no significant difference.}
\end{frame}
\begin{frame}{Geometry-aware initialization is useful when the geometry persists}

\begin{enumerate}
\item \textbf{Measure the target site.} Attention outputs often concentrate strongly; residual streams may not.
\item \textbf{Choose directions from data.} ASI improves the starting dictionary, with a one-time SVD/projection cost.
\item \textbf{Track three outcomes.} Feature survival, reconstruction, and downstream LM loss need separate checks.
\item \textbf{Respect the training regime.} High initial $L_0$ can erase gains; very large dictionaries benefit from SparseAdam.
\end{enumerate}
\medskip
\alert{Open question:} How do low-rank geometry and feature competition jointly produce dead features?

\papersource{Paper, Sec. 6, PDF p. 8; App. J--N. Final interpretation based on the reported controls.}
\end{frame}
\begin{frame}{References and implementation entry points}

\begin{itemize}
\item Wang, Ge, Shu, He, and Qiu.\\
\emph{Dimensional Collapse in Transformer Attention Outputs: A Challenge for Sparse Dictionary Learning}.\\
\href{https://arxiv.org/abs/2508.16929v4}{arXiv:2508.16929v4}, 11 February 2026.
\item \href{https://proceedings.mlr.press/v306/wang26bh.html}{ICML 2026 proceedings}, PMLR 306:126304--126331.
\item \href{https://github.com/OpenMOSS/Llamascopium}{OpenMOSS/Llamascopium}: official public implementation.\\
\href{https://github.com/OpenMOSS/Llamascopium/blob/dev/src/llamascopium/initializer.py}{Initializer}: active-subspace switch and dimension configuration.
\end{itemize}
\medskip
Figures are reused from the authors' arXiv v4 source or cropped from its PDF. Calculated comparisons are labeled explicitly.

\papersource{Publication and code checked 11 November 2026. This deck is a reading seminar, not an author presentation.}
\end{frame}
\appendix
\begin{frame}{Backup: ASI in row-vector pseudocode}

\begin{align*}
\widetilde X&=X-\operatorname{mean}_{\rm tokens}(X),\\
U,\Sigma,V^\top&=\operatorname{SVD}(\widetilde X),\\
B&=V_{:,:d_{\rm init}},\\
W_D&\leftarrow (W_DB)B^\top,\\
W_E&\leftarrow W_D^\top.
\end{align*}
\begin{itemize}
\item $X$: $n\times d$; $B$: $d\times d_{\rm init}$; $W_D$: $h\times d$.
\item The factored update avoids materializing the $d\times d$ projection.
\item Subsequent SGD/Adam training is unrestricted.
\end{itemize}

\papersource{Paper, Sec. 5.2/App. K, PDF pp. 6, 22. Variable-name and prose inconsistencies resolved using the matrix update.}
\end{frame}
\begin{frame}{Backup: gains persist across the tested sparsity frontier}

\fig{l0}{3.7cm}
\begin{itemize}
\item Fixed 32,768-feature dictionary, $L_0\in\{40,50,60,70,80\}$.
\item ASI has the best mean values for all three metrics in Table 4.
\item Increasing $L_0$ reduces base TopK death and narrows reconstruction gaps.
\item Separate 15-seed test: ASI vs. AuxK $p=1.27\times10^{-6}$ for $\Delta$ LM Loss.
\end{itemize}

\papersource{Paper, Fig. 7, PDF p. 7; Tables 4--5, PDF p. 20. Test: one-tailed Welch, $k=50$.}
\end{frame}
\begin{frame}{Backup: a near-full-rank site offers little room for ASI}

\fig{d_init_resid}{3.9cm}
\begin{itemize}
\item Target: post-layer-15 residual stream of Llama-3.1-8B.
\item ASI provides minimal gains relative to ordinary initialization.
\item Over-restricting directions damages reconstruction; assess the spectrum first.
\end{itemize}

\papersource{Paper, App. J.2 and Fig. 13, PDF p. 22. These curves describe a different activation site from the main experiment.}
\end{frame}
\begin{frame}{Backup: evaluation conventions determine the interpretation}

\begin{itemize}
\item \textbf{Spectra:} 10M activations per site; mean-centered; exclude extreme activation norms. Five samples give effective rank $2523.165\pm0.404$ in App. C.
\item \textbf{Dead features:} no activation in 10M evaluation tokens.
\item \textbf{NMSE:} retain the paper's reported metric; v4 does not spell out its normalization denominator.
\item \textbf{$\Delta$ LM Loss:} positive degradation, lower is better in tables; interpreted here as $L_{\rm SAE}-L_{\rm original}$.
\item \textbf{Optimizer comparison:} learning rates swept separately; one-time SVD is additional preprocessing.
\end{itemize}

\papersource{Paper, Secs. 2.3, 4.2, 5.2; App. C/E, PDF pp. 3--6, 14, 16--18.}
\end{frame}
\end{document}