A is for Absorption: Studying Feature Splitting and Absorption in Sparse Autoencoders

## Abstract

Abstract Sparse Autoencoders (SAEs) have emerged as a promising approach to decompose the activations of Large Language Models (LLMs) into human-interpretable latents. In this paper, we pose two questions. First, to what extent do SAEs extract monosemantic and interpretable latents? Second, to what extent does varying the sparsity or the size of the SAE affect monosemanticity / interpretability? By investigating these questions in the context of a simple first-letter identification task where we have complete access to ground truth labels for all tokens in the vocabulary, we are able to provide more detail than prior investigations. Critically, we identify a problematic form of feature-splitting we call “feature absorption” where seemingly monosemantic latents fail to fire in cases where they clearly should. Our investigation suggests that varying SAE size or sparsity is insufficient to solve this issue, and that there are deeper conceptual issues in need of resolution. We release a feature absorption explorer at https://feature-absorption.streamlit.app .

## 1 Introduction

Figure: Figure 1: In feature absorption, an SAE latent appears interpretable, but has arbitrary exceptions where token-aligned latents “absorb” the feature direction and activate instead.
Refer to caption: x1.png

Large Language Models (LLMs) have achieved remarkable performance across a wide range of tasks, yet our understanding of their internal mechanisms lags behind their capabilities. This gap between performance and interpretability raises concerns about the “black box” nature of these models (Rudin, ). The field of mechanistic interpretability aims to address this issue by reverse-engineering the internal algorithms of neural networks and performing causal analysis on them (Olah et al., ).

One recent promising approach in this field is the use of Sparse Autoencoders (SAEs), which have shown potential in decomposing the dense, polysemantic activations of LLMs into more “interpretable” latent features (Huben et al., ; Bricken et al., ) using sparse dictionary learning (Olshausen & Field, ). SAE neurons (hereafter called “latents”) (^1^11We use latents to prevent overloading the term feature, which we reserve for human-interpretable concepts the SAE may capture. This breaks from earlier usage which used feature for both (Elhage et al., ), but aligns with the terminology in (Lieberum et al., ) and makes the distinction more clear.) are said to be interpretable if they appear to detect some property of the input (which we refer to as a “feature”) and classify that feature with high precision / recall (Bricken et al., ).

However, despite these theoretical advantages, most existing work on SAE interpretability mainly studies max activating examples (Huben et al., ), which may be misleading. There are more rigorous works which only measure precision (Bricken et al., ; Templeton et al., ; Kissane et al., ). Recent work has briefly explored recall and found it to be worse than naively expected, but this remains poorly understood (Olah et al., ). We build on this work by evaluating precision / recall on a large number of Gemma Scope SAEs, and offer a partial explanation for lower-than-expected recall of SAE latents in the form of “feature absorption”.

Our key contributions in this investigation include the following:

- 1.
We identify numerous SAE latents appearing to classify first-letter features. We calculate their precision / recall on the first letter identification tasks (as a proxy for monosemanticity / interpretability) and find they significantly underperform linear probes.
- 2.
We find that latents which seem outwardly to classify the same feature can have vastly different precision / recall and that this tradeoff is mediated by various factors, mainly sparsity and width of the SAE.
- 3.
Most importantly, we identify and quantify a variant of feature-splitting we call “feature absorption”, where an SAE latent appears to track a human-interpretable concept, but it fails to activate on seemingly arbitrary tokens. Instead, approximately token-aligned latents activate and contribute a portion of the probe direction, “absorbing” the feature. This is described in Figure 1.

We believe that feature absorption poses an obstacle to the practical application of SAEs since it suggests SAE latents may be inherently unreliable classifiers. This is particularly important if we seek to use them in safety applications where we need confidence that latents are fully tracking behaviours, such as bias or deceptive behavior. Furthermore, techniques which seek to describe circuits in terms of a sparse combination of latents with also be more difficult in the context of feature absorption (Marks et al., ).

Our code is publicly available on Github (^2^22Code for experiments is available at [https://github.com/lasr-spelling/sae-spelling](https://github.com/lasr-spelling/sae-spelling)).

## 2 Background

#### Linear probing:

A linear probe is a simple linear classifier trained on the hidden activations of a neural network, typically using logistic regression (LR) (Alain & Bengio, ).

#### K-sparse probing:

A k-sparse probe (Gurnee et al., ) is a linear probe trained on a sparse subset of $k$ neurons or SAE latents. Training a k-sparse probe first requires selecting the $k$ best neurons or SAE latents that in-aggregate act as a good classifier, and then training a standard linear probe on just those $k$ neurons or latents.

Gurnee et al. () proposed several methods of estimating the best $k$ neurons or features to pick, one of which involves first training a LR probe with a L1 loss term, and selecting the $k$ largest elements by probe weight. When we refer to k-sparse probing in this work, we use this method of selecting $k$ features.

#### Sparse autoencoders:

An SAE consists of an encoder, $W_{enc}$, a decoder, $W_{dec}$, and corresponding biases $b_{enc}$ and $b_{dec}$. The SAE has a nonlinearity, typically a ReLU (or variants such as JumpReLU (Rajamanoharan et al., )). Given input activation, $a$, the SAE computes a hidden representation, $f$, and reconstruction, $\hat{a}$:

$$ $\displaystyle f=$ $\displaystyle ReLU(W_{enc}a+b_{enc})$ (1) $\displaystyle\hat{a}=$ $\displaystyle W_{dec}f+b_{dec}$ (2) $$

SAEs attempt to reconstruct input activations by projecting into an overcomplete basis using a sparsity-inducing loss term (typically $L1$ loss), or a certain number of non-zero features ($L0$) on the hidden activations. SAEs learn feature decompositions in an unsupervised manner, and while the sparsity penalty is meant to encourage monosemantic features, it is often hard to judge if the features learned are interpretable or to say with certainty that the features the SAE learned are faithful to the computation performed by the underlying LLM.

#### SAE feature ablation:

We often want to understand how an SAE latent causally influences a downstream output. In an ablation study, the latent in question is removed from the computation graph of the model to see the effect this has on a downstream metric. A negative ablation effect means removing the SAE latent would lower the metric.

We follow the work of Marks et al. () and provide the procedure in Algorithm 1 below. We also make use of the integrated-gradients (IG) approximation (Sundararajan et al., ) to improve the speed of running multiple ablation experiments.

Figure: Algorithm 1 SAE Latent Ablation

## 3 Experimental setup

Our experiments focused on predicting the first-letter of a single token containing characters from the English alphabet (a-z, A-Z) and an optional leading space. We use in-context learning (ICL) prompts to elicit knowledge from the model, using templates of the form:

{token} has the first letter: {capitalized_first_letter}

An example of an ICL prompt consisting of 2 in-context examples is shown below. The model should output the _D token:

In the above prompt, we extract residual stream activations at the _dog token index. These activations are used both for LR probe training and for applying SAEs. We use a train/test split of 80% / 20%, and evaluate only on the test set of the probes, including when running experiments on SAEs. When applying SAEs, we always include the SAE error term to avoid changing model output.

To determine the causal effect of SAE latents on the first-letter identification task we conduct ablation studies. We use a metric consisting of the logit of the correct letter minus the mean logit of all incorrect letters. This measures the propensity of the model to choose the correct starting letter as opposed to other letters. Formally, our metric $m$ is defined below, where $g$ refers to the final token logits, $L$ is the set of uppercase letters, and $y$ is the uppercase letter that is the correct starting letter:

$$ $m=g[y]-\frac{1}{|L|-1}\sum_{l\in\{L\setminus y\}}g[l]$ $$

To determine how well multiple features perform as a classifier when used together, we use k-sparse probing, increasing the value of $k$ from $1$ to $15$. We train a LR probe using a L1 loss term with coefficient $0.01$, and select the top $k$ features by magnitude.

We use the base Gemma-2-2B model for all of our studies, along with the full set of Gemma Scope residual stream SAEs of width 16k and 65k released by Deepmind (Lieberum et al., ).

## 4 Results

Our results are divided into three sections. First, we compare the performance of linear probes with SAE latents on recovering first-character information from model activations, showing that despite appearing to track first letter features, a wide variety of precision / recall is achieved. Second, we motivate our definition of feature absorption with a case-study, emphasizing how an absorbing feature can unexpectedly causally mediate first letter information whilst the first-letter latent (unexpectedly) fails to fire. Finally, we attempt to quantify feature splitting and feature absorption, showing that tuning of hyper-parameters may partially assist but not fully alleviate feature absorption.

### 4.1 Do SAEs learn latents that track first letter information?

We compare the performance of LR probes with the performance of the SAE latent whose encoder direction has highest cosine similarity with the probe, resulting in 26 “first-letter” latents. We observed that for each probe, there was clearly one or at most a couple of outlier SAE latents with high probe cosine similarity. Full plots of cosine similarity vs letter are shown in Appendix A.5.

We also experimented with using k=1 sparse probing to identify SAE latents (Gao et al., ), and find this gives similar results. Further comparison of k=1 sparse probing and encoder cosine similarity is explored in Appendix A.4.

Figure: (a) Mean F1 score using top SAE encoder feature by layer for Gemma Scope SAEs and LR probe.
Refer to caption: x2.png

Figure: (a) Mean precision using top SAE encoder feature vs L0 for all Gemma Scope SAEs.
Refer to caption: x5.png

We observe wide variance in the performance of Gemma Scope SAEs at the first-letter identification task, but no SAE matches LR probe performance. We show the mean F1 score by layer as well as the F1 score of the LR probe in Figure 2(a). We further investigate the F1 score of these SAE encoder latents as a function of L0 and SAE width in Figures 2(b) and 2(c).

Whether or not an SAE learns a clear “first-letter” latent for each letter is highly dependent on L0, with low L0 SAEs tending to learn high-precision low-recall latents, and high L0 SAEs learning low-precision high-recall latents (Figure 3). We caution drawing conclusions about an “optimal” L0 from these plots, as we find further variance when broken-down by letter, shown in Appendix A.5.

### 4.2 Why do SAE latents underperform?

The Gemma Scope layer 3, 16k width, 59 L0 SAE has a latent, 6510, which appears to act as a classifier for “starts with S”, achieving an F1 of 0.81. However, this latent fails to activate on some tokens the probe can classify, and which the model can spell, such as the token _short.

Figure 4(a) shows a sample prompt containing a series of tokens that start with “S”, and the activations of top SAE latents by ablation score for these tokens. The main “starts with S” latent, 6510, activates on all these tokens except _short. This SAE also has a token-aligned latent, 1085, which activates on variants of the word “short” (“ short”, “SHORT”, etc…). The Neuronpedia dashboard (Lin & Bloom, ) for feature 1085 is shown in Appendix A.8. For the token _short, the main “starts with S” latent does not activate but the “short” latent activates instead.

Figure: (a) Layer 3, L0=59 SAE feature activations for tokens that start with “S”. The core “starts with S” feature, 6510, fails to activate on the token _short. The “short”-token aligned feature 1085 activates instead.
Refer to caption: x8.png

Latent 1085 has a cosine similarity with the “starts with S” probe of $0.12$, indicating it contains a component of the “starts with S” direction, although much smaller than the main “starts with S” latent. Cosine similarity of the SAE decoder with the “starts with S” LR probe is shown in Figure 4(b). Interestingly, despite latent 1085 having only about $1/5$ the cosine similarity with the probe as the main latent 6510, we see it activates with about $5$ times the magnitude of latent 6510 on the _short token, thus contributing a similar amount of the “starts with S” probe direction to the residual stream.

We conduct an ablation experiment on the _short token, shown in Figure 5(a), and see that latent 1085 has a dramatically larger ablation effect compared with all other SAE features. This suggests latent 1085 is causally responsible for the model knowing that _short starts with S.

Is it possible that the probe projection is not the causally important component of feature 1085? We conduct another ablation experiment, except now we remove the probe direction from feature 1085 via projection before ablation. The results of this ablation experiment are shown in Figure 5(b). After removing the probe component from feature 1085, it no longer has a significant ablation effect. Thus we know the probe projection of feature 1085 is responsible for model behavior.

These experiments show the “starts with S” feature has been “absorbed” by the token-aligned latent 1085, likely along with other semantic concepts related to the word “short”. After observing that the main “starts with S” latent 6510 activates on most tokens that begin with “S”, it may be tempting to conclude this latent tracks the interpretable feature of beginning with the letter “S”. However, this latent quietly fails to activate on the _short token, leading us to a false sense of understanding.

We call this phenomenon feature absorption. In feature absorption a seemingly interpretable SAE latent fails to activate on arbitrary positive examples, and instead the feature is “absorbed” into approximately token-aligned latents.

Feature absorption is likely a logical consequence of SAE sparsity loss. If a dense and sparse feature co-occur, absorbing the dense feature into a latent tracking the sparse feature will increase sparsity.

Figure: (a) Ablation effect for _short token, indicating that feature 1085, is responsible for the “starts with S” concept for the _short token. The main “starts with S” latent, 6510, does not activate on the _short token.
Refer to caption: x10.png

### 4.3 Measuring feature splitting and feature absorption

#### Feature splitting

A key phenomenon identified from previous studies of SAEs is feature-splitting (Bricken et al., ), where a feature represented in a single latent in a smaller SAE can split into two or more latents in a larger SAE. During our experiments, we found strong evidence of feature-splitting in the Gemma Scope SAEs.

For instance, in the layer 0, 16k width, 105 L0 SAE, we find two encoder latents (id:7112 and id:7657 (^3^33[https://www.neuronpedia.org/list/cm0h1n2mt00019jdk274owq9e](https://www.neuronpedia.org/list/cm0h1n2mt00019jdk274owq9e))) which align with the “L” starting letter probe. Inspecting max activating examples, we see latent 7112 activates on tokens starting with lowercase “l”, while 7657 activates on tokens starting with uppercase “L”. Some activating examples for these latents are shown in Table 1.

**Table 1: Sample max activating examples for latents 7112 and 7657 for Gemma Scope 16k, layer 0, 105 L0 from Neuronpedia. The token where the SAE feature activates is highlighted in green. Latent 7112 appears to be a lowercase “L” starting-letter latent, and latent 7657 appears to be a corresponding uppercase “L” latent.**
| Latent 7112 | Latent 7657 |
| --- | --- |
| žda se naplaćuje naknada | LC, an aluminum boat |
| . E. Søli, 20 | as LIFT and LF-Net. Once |
| a></code></li></ul | latter’s sister Louise, who in |

Feature splitting like this is not necessarily problematic for interpretability efforts since the split features are still easily identifiable, and depending on the context it may be more useful to have either a single “starts with L” latent or a pair of “starts with uppercase / lowercase L” latents.

We measure feature splitting using k-sparse probing (Gurnee et al., ) on SAE activations. If increasing the k-sparse probe from $k$ to $k+1$ causes a significant increase in probe F1 score, then the additional SAE latent provides a meaningful signal, and the combination of these $k+1$ latents is likely a feature split. In the example of the uppercase “L” and lowercase “l” split, a k-sparse probe with $k=2$ trained on both these these features should predict “starts with letter L” much better than either feature on its own. Figure 6(a) shows F1 vs K for letters “L” and “N”. The “L” k-sparse probe shows a significant jump in F1 score moving from k=1 to k=2 corresponding to feature splitting, while the F1 score for the “N” k-sparse probe is relatively constant.

Figure: (a) K-sparse probing results for letters “L” and “N”, layer 0, 16k width, 105 L0. “L” shows a significant improvement in F1 between k=1 and k=2 corresponding to feature splitting.
Refer to caption: x12.png

We detect feature splitting by measuring whether increasing $k$ by one causes a jump in F1 score by more than threshold $\tau$. We set $\tau=0.03$ after manually inspecting latents with various thresholds. Figure 6(b) shows feature splitting vs L0 for all 16k and 65k width Gemma Scope SAEs.

#### Feature absorption

The single latent or a set of traditional feature split latents that seem to act as a classifier for a human-interpretable feature like “starts with S” fail to fire in a seemingly arbitrary number of cases. What fires instead are approximately token-aligned latents with small but positive alignment with the LR probe. We say these latents are absorbing the feature.

Figure: (a) Mean feature absorption rate vs layer on the first-letter task, Gemma Scope 16k and 65k SAEs. We do not see an obvious pattern in absorption rates by layer.
Refer to caption: x14.png

We quantify the extent to which feature absorption occurs with the metric feature absorption rate. We first find $k$ feature splits for a first-letter feature using a k-sparse probe. We then find false-negative tokens that all $k$ feature-split SAE latents fail to activate on, but which the LR probe correctly classifies, and run an integrated-gradients ablation experiment on those tokens. The ablation effect finds the most causally important SAE latents for the spelling of that token. If the SAE latent receiving the largest negative magnitude ablation effect has a cosine similarity with the LR probe above $0.025$, and is at least $1.0$ larger than the latent with the second highest ablation effect, we say that feature absorption has occurred. These thresholds were chosen from manual inspection of the data. We then calculate feature absorption rate as below:

$$ $\mathtt{absorption_rate}=\frac{\mathtt{num_absorptions}}{\mathtt{lr_probe_ true_positives}}$ $$

If there are more than 200 false negative per letter, we randomly sample 200 samples to estimate the number of absorptions. We see that absorption rate increases with higher sparsity and higher SAE width. Lower L0 likely pushes the SAE to absorb dense features like spelling information across multiple latents, thereby increasing feature sparsity. Feature absorption rate vs L0 for Gemma Scope SAEs layers 0-17 is shown in Figure 7(b). Absorption rate by letter is shown in Appendix A.7.

Our metric cannot capture absorption past layer 17 in Gemma 2 2B since we rely on ablation experiments to be certain the absorbed feature causally mediates model behavior. Past layer 17, attention has already moved the starting letter information from the source token into the final token position, so any ablations on the source token past layer 17 have little effect. This is a limitation of our absorption metric - we rely on ablation to be certain of the causal impact of absorbed features on model behavior, but this limits the layer depth our metric can be applied. We discuss this further in Appendix A.6.

Our absorption metric is not perfect, and is likely an under-estimate of the true level of feature absorption. We only consider absorption to have occurred if a single SAE latent has a much larger ablation effect than all other latents, and if the main SAE latents for a feature do not activate at all. Our metric will not capture multiple absorbing latents activating together, or the main latents activating but very weakly. Regardless, we feel our metric is a reasonable conservative baseline.

## 5 Related work

### 5.1 Applications of Probes and SAEs for Model Interpretability

Probing methods have often been used to extract interpretable information from language models. However, the existence of such a representation does not guarantee that the model relies on this representation in its computation graph (Elazar et al., ).

Prior work has shown that many human-interpretable concepts in LLM activations are represented as linear directions in activation space, known as the linear representation hypothesis (Elhage et al., ; Park et al., ). Li et al. () used non-linear probes to recover board representations from a transformer trained on Othello scripts (“OthelloGPT”). However, Nanda () later showed that linear representations were not only recoverable but also editable.

Recent work has focused on applying SAEs to extract human-interpretable explanations of model internals. Karvonen et al. () investigated how SAEs represent board states of Chess and Othello, and attempted to develop a more objective way of evaluating SAEs. However, it is difficult to apply this to existing SAEs trained on real LLM activations.

Other work has noted poor recall / precision of SAE latents compared to known proxies (Olah et al., ; Kissane et al., ; Templeton et al., ). We build on this work by evaluating a large number of Gemma Scope SAEs to demonstrate how precision / recall is mediated by sparsity, and also offer a possible explanation of low recall due to feature absorption.

### 5.2 Character-level information in language models

The ability of LLMs to learn character-level information from ostensibly character-blind tokens has been studied by various scholars, though no clear mechanism has yet been established. Kaushal & Mahowald () trained MLPs from the embedding layers of GPT-J as probes for each letter in the alphabet, again finding good performance that implied character-level information was represented, but did not look into the model internals to explain how these representations were being used. In a follow-up work, Watkins & Bloom () demonstrated that even linear probes on the embedding layers perform comparably well to MLPs in extracting character-level information.

In contrast to the above approaches, we train various probes for each character on multiple layers, and compare with SAE latents for a variety of layers.

### 5.3 Decomposing SAE Latents

The phenomenon of SAEs of different sides splitting a feature into various smaller latents was first described in Bricken et al. (), which noted that different SAE widths and sparsities induce latents of different granularity, with wider SAEs often learning more specific variants of features.

Bussmann et al. () find that by training an SAE on the decoder or another SAE, a technique called Meta-SAEs, it is possible to break down a single SAE latent like “Einstein” into subcomponents like “German” and “Physicist” and “starts with E”. Meta-SAEs may provide a promising future research direction to overcome the feature absorption phenomenon we describe in this paper.

## 6 Discussion

#### Interpretability of SAE latents

In this work, we use a simple first-letter identification task to investigate whether SAEs extract monosemantic and intrepretable features, and how this is affected by varying hyperparameters like SAE size, sparsity, or layer. We find that the SAE latents we investigated were not interpretable and that varying the sparsity or the size of the SAE did not meaningfully change this.

One may argue it is unreasonable to judge an SAE latent against a linear probe directly optimized to perform the same classification task and that it has been established that sometimes unsupervised methods can surprise us (Nanda, ). However, we argue that for a latent to be considered interpretable, its behavior should match what one would reasonably expect the latent to be doing after inspecting its activation patterns. In our experiments, we use SAE latents that do appear to perform first letter classification, and then evaluate how well they perform this task. We validate as well that these latents causally mediated model performance on the first-letter task in Appendix A.3. We are convinced that these latents should reasonably be considered “first-letter” latents and that their performance on the first-letter identification task is a valid measure of their interpretability.

#### Feature absorption

In trying to understand why SAE latents fail to match the performance of LR probes, we identified a form of feature splitting we call “feature absorption”. Feature absorption may be particularly problematic for SAEs because it creates an interpretability illusion where we believe we have found an interpretable latent, but absorption induces lower recall by creating clear false negatives to the mainline interpretation of the latent. For example, we may believe we have found a SAE latent which tracks deceptive behavior in the model, but due to feature absorption, there may be many cases where that latent fails to fire. This lower recall poses problems for methods which rely on using SAEs to find sparse circuits (Marks et al., ), as the number of latents needed to characterize model behavior may be much larger than expected. We find that feature absorption happens even in high-recall latents, so this is not only a problem for low L0 SAEs and appears to be a more fundamental issue.

Feature absorption is not predicted by toy-models where features do not co-occur (Bricken et al., ). While more work is required to understand why feature absorption occurs, we suspect it is a consequence of co-occurrence between sparse and dense features. If a dense feature like “starts with letter D” always co-occurs with a more sparse feature like “dogs”, the SAE can increase sparsity by absorbing the “starts with D” feature into a “dogs” latent.

It remains to be seen if we can predict or identify excepted instances where a feature “should have activated” but does not activate due to absorption. One promising direction is meta-SAEs, a novel method for decomposing SAE latents and may decompose absorbed features (Bussmann et al., ). One interpretation of our results is that competition may exist between “latents” and “meta-latents” for activation on particular examples and that re-allocation of examples between SAE latents enables SAEs to interpolate between different possible decompositions.

#### Future Work

A primary goal of future work should be to secure further external validity of our findings. This could include finding examples of feature absorption in SAEs trained on other models, with other architectures, or finding examples of feature absorption unrelated to character identification. We expect it should be possible to demonstrate feature absorption in a toy model by mixing dense features with sparse features that always co-occur with these dense features.

We hope as well this investigation may lead to research into solutions, particularly those involving Meta-SAEs (Bussmann et al., ), to solve or mitigate feature absorption. Anoter possible solution may be attribution dictionary learning (Olah et al., ).

#### Limitations

The results presented in our paper are based on a single model (Gemma-2-2B) using one type of SAE architecture (JumpReLU). Our feature absorption metric requires having ground-truth knowledge of true labels to first train a LR probe, whereas many features of interest in a LLM lack such clear-cut ground-truth labels. Our metric uses ablation effect to ensure absorbed features causally mediate model behavior, but therefore cannot be easily used in final model layers.

## 7 Acknowledgements

This project was produced as part of the LASR Labs research program.

## Appendix A Appendix

### A.1 Glossary of Terms

#### Sparse Autoencoders (SAEs):

Neural networks trained to reconstruct their input while enforcing sparsity in their hidden layer. In the context of this paper, SAEs are used to decompose the dense activations of language models into more interpretable features.

#### SAE error term:

When inserting a SAE into the computation path of the model, errors in SAE reconstruction will propagate to later parts of the model and can change the model output. We refer to the error as the SAE error term, and corresponds to the difference between the SAE output and the original SAE input activation. Marks et al. () introduced the idea of adding this error term back to the SAE output to ensure that the SAE does not change model output.

#### Latent:

We refer to neurons in the hidden layer of a SAE as latents to avoid overloading the term “feature”. This is in contrast to earlier work which used the term “feature” to refer to both human-interpretable concepts and SAE hidden layer neurons.

#### Feature:

We use the term “feature” to refer to an idealized human-interpretable concept that the model represent in its activations and which a SAE latent may or may not represent.

#### Monosemantic:

Referring to a feature or representation that corresponds to a single, clear semantic concept. In the context of SAEs, a monosemantic feature would ideally capture one interpretable aspect of the input.

#### Interpretable:

A latent being interpretable is not well defined in the field, making it difficult to ensure that different authors mean the same thing when referring to SAE interpretability. When we refer to a SAE latent as being interpretable in this work, we mean that it should behave in line with how it appears to behave after inspecting its activation patterns. If an SAE latent appears to track a feature X by a reasonable inspection of its activations but has subtle deviations from this behavior in reality, we say this is not interpretable. We thus measure interpretability via classification performance when a latent appears to be a classifier over some feature.

#### Feature dashboard:

A dashboard showing activation patterns and max-activating examples for a SAE latent. Feature dashboards are commonly used to interpret the behavior of an SAE latent.

#### Neuronpedia:

A platform, [https://neuronpedia.org](https://neuronpedia.org), which hosts feature dashboards for popular SAEs (Lin & Bloom, ).

#### Feature splitting:

A phenomenon in SAEs introduced by Bricken et al. (), where a SAE latent tracks a general feature in a narrow SAE, but splits into multiple more specific SAE latents in a wider SAE. For instance, a latent tracking “starts with L” in a narrow SAE may split into a latent tracking “starts with capital L” and a latent tracking “starts with lowercase L” in a wider SAE.

#### Feature absorption:

A problematic form of feature splitting where a SAE latent appears to track an interpretable feature, but that latent has seemingly arbitrary exception cases where it fails to fire. Instead, an approximately token-aligned feature “absorbs” the feature direction and fires in place of the main latent.

#### Circuit:

In the context of neural network interpretability, a circuit refers to a subgraph of neurons or features within a neural network that work together to perform a specific function or computation. The study of circuits aims to understand how different components of a neural network interact to process information and produce outputs.

#### Linear probe:

A simple linear classifier (typically logistic regression) trained on the hidden activations of a neural network to predict some property or task. Used to assess what information is linearly decodable from the network’s representations.

#### K-sparse probing:

A variant of linear probing where only the k most important features (as determined by some selection method) are used to train the probe. This helps identify which specific neurons or features are most relevant for a given task.

#### Ablation study:

An experimental method where a component of a system (in this case, a neuron or feature in a neural network) is removed or altered to observe its effect on the system’s performance. This helps determine the causal importance of the component.

#### Integrated gradients (IG):

An attribution method that assigns importance scores to input features by accumulating gradients along a path from a baseline input to the actual input. In this paper, it’s used as an approximation technique for ablation studies.

#### In-context learning (ICL):

A paradigm where a language model is given examples of a task within its input prompt, allowing it to adapt to new tasks without fine-tuning. Often used with few-shot learning techniques.

#### Residual stream:

In the context of transformer architectures, the residual stream refers to the main information flow that bypasses the self-attention and feed-forward layers through residual connections.

#### Logits:

The raw, unnormalized outputs of a neural network’s final layer, before any activation function (like softmax) is applied. In language models, logits typically represent the model’s scores for each token in the vocabulary.

#### Activation patching:

An interpretability technique where activations at specific locations in a neural network are replaced or modified to observe the effect on the network’s output. This helps in understanding the causal role of different parts of the network in producing its final output.

### A.2 How good is Gemma-2 on character identification tasks?

Figure: Figure 8: Baseline performance for Gemma-2-2B on first-letter identification and full-token spelling by token length.
Refer to caption: x16.png

We evaluate how well can Gemma-2-2B identify the first letter or all the letters in a token (spelling the full token). We evaluate the accuracy of the model on all tokens in the LR probe validation set with a prompt containing 10 in-context examples selected at random from the full vocabulary. Our results are shown in Figure 8.

We see that performance on the first-letter identification task is high throughout token length, while the full-word spelling performance decreases as the length of the token increases.

### A.3 Intervening on the first letter

If the model is using the identified SAE latents for predicting the first letter we should also be able to change what first letter it predicts just by changing the activations. For this experiment we use the SAE latents most cosine similar with the LR probe for the true first letter and for a new randomly selected letter. We take the intermediate activations of Gemma-2-2B in the residual stream and encode them using the SAE. Then we zero out the activation of the SAE latent associated with the original letter and change the activation of the SAE latent associated with the new letter into the average activation it has on tokens starting with this new letter.

Editing works better with latents from the narrower 16k SAE compared to the 65k, with the best L0s in the 75-150 range. This corresponds to the observed pattern of these SAE latents having higher F1 scores for classification. We report the results in Figure 9. The best SAEs on the layers 7-9 can achieve a substantial replacement, but note that the averages hide variance across individual tokens, where some get edited completely and others get unaffected. The edit success also varies based on the true first letter and the random new letter; for illustration we show a breakdown by letter for two specific SAEs in layer 7 in Figure 10.

Figure: (a) Comparing success in editing out the true first letter and making the model predict a randomly selected new letter across layers 0-9 for all 16k and 65k Gemma Scope SAEs.
Refer to caption: x17.png

Figure: Figure 10: Comparing the edit success broken down by the letter at layer 7 for two SAEs; SAE width 16,000 and L0 of 137 and SAE width 65,000 and L0 of 107. For each original letter we draw a sample of 100 tokens and average the decrease in probability of the correct first letter and increase in probability of a new random letter.
Refer to caption: x19.png

### A.4 Probe cosine similarity vs k=1 sparse probing

The first step when searching for a SAE feature that acts as a first-letter classifier involves searching for SAE feature which best acts as a classifier. In Figure 2, we achieve this by first training a LR probe on the first-letter task and using cosine similarity between that probe and the SAE encoder to find the best feature for the first-letter task. We also investigated using k-sparse probing with k=1 to select the best SAE feature instead. This involves training a linear probe with L1 loss and selecting the feature with the highest positive weight from the probe.

We find that both k=1 sparse probing yield nearly identical results, as seen in Figures 11 and 12. Additionally Figure 13 shows the cosine similarity of the LR probe with each SAE feature by letter for the canonical Gemma Scope layer 0 16k width SAE. In most cases there is an obvious probe-aligned feature. Likely any reasonable method of feature selection will find the same feature for these cases. We thus decided to use cosine similarity between the SAE encoder and a LR probe as our selection criteria for single SAE features as this is a simpler metric and less computationally intensive to compute.

Figure: (a) Mean F1 score on first-letter classification task using top SAE encoder feature by cosine similarity with the LR probe vs L0 for all Gemma Scope SAEs layers 0-9.
Refer to caption: x20.png

Figure: (a) Mean F1 score on first-letter classification task using top SAE encoder feature by cosine similarity with the LR probe vs layer for all Gemma Scope SAEs layers 0-9.
Refer to caption: x22.png

Figure: Figure 13: Decoder cosine similarities with the LR probe by letter, Gemma Scope 16k layer 0 l0=105. Most letters have one or two obvious SAE features which align with the probe.
Refer to caption: x24.png

### A.5 Precision, recall, and F1 score for the first-letter task

We evaluated precision, recall, and F1 score for the first-letter classification task, and found that the precision and recall vary depending on the L0 of the SAE. Low L0 SAEs learn high precision, low recall features, while high L0 SAEs learn low precision, high recall features. These results are shown in Figure 14. We thus chose to use F1 score as our core metric in this paper to balance precision and recall as many if the SAEs we tested have extreme values in either precision or recall.

Figure: (a) Mean precision on first-letter classification task vs L0 for all Gemma Scope SAEs layers 0-9. Features are selected via k=1 sparse probing
Refer to caption: x25.png

While it may appear that there is an optimal L0 from looking at aggregate statistics across letter, we find that breaking down the F1 vs L0 plot by letter reveals that the optimal L0 appears different for different letters, with low frequency letters like z actually having the best F1 score at the lowest L0, while other letters instead have an optimal L0 around 30-50. Figure 15 shows these results broken down by letter.

Figure: Figure 15: F1 vs L0 by letter. SAE features are picked using k=1 sparse probing.
Refer to caption: x27.png

### A.6 Causal interventions and absorption

In this work, we rely on causal interventions like ablation experiments to verify that SAE latents have a causal impact on model behavior. In these experiments for spelling tasks, we set up an ICL prompt to elicit spelling information from the model, for instance the ICL prompt below:

In this ICL prompt, we would apply an SAE and train LR probes on the _dog token position, and expect that the model will output the token _D. When we intervene on the _dog, we can track the causal changes to model outputs by applying a metric to the output logits, e.g. checking how our intervention increases or decreases the _D logit relative to other letters.

We use these interventions as part of our absorption metric to ensure that when we claim that “absorption” is occurring, we verify that the absorbing feature has a causal impact on model outputs. This is stronger evidence than only noting a cosine similarity between the absorbing feature, but this means that our absorption metric cannot classify absorption at later model layers.

During a LLM forward pass, the model first collects relevant information on a token in that token position, and attention heads then move relevant information from earlier tokens to later tokens (Geva et al., ; Meng et al., ). If we assess ablation effect at layers after which model attention has already pulled relevant information from the subject tokens into the final output token, the ablation effect will be 0. For Gemma 2 2B on the first-letter spelling task, we find this movement of first-letter spelling information occurs around layer 18.

Figure 16 shows an activation patching experiment (Meng et al., ) on a sample first-letter spelling prompt. In this experiment, we see that near layer 18 the model moves first-letter spelling information from the subject token to the prediction token.

Figure: Figure 16: Residual stream attribution patching for a sample first-letter spelling prompt, Gemma 2 2B. After around layer 18, model attention moves the relevant spelling information from the source token to the prediction location.
Refer to caption: x28.png

As a result, our feature absorption metric will not function past layer 18 in Gemma 2 2B, and we thus focus on layers 0-17 for our analysis of feature absorption. We believe that feature absorption is still occurring in SAEs past layer 18, but we lose the ability to make causal claims that the absorbing features are used by the model to make predictions. Given that this paper is trying to highlight the existence of feature absorption, we felt it is more important to have a metric which is robust and has the backing of causal analysis but which cannot be used at all model layers. Future work may make a different trade-off and choose a feature absorption metric which can work at all model layers, for instance relying only on cosine similarity between absorbing features and a LR probe to determine absorption.

### A.7 Additional plots

In this section, we include additional plots that are too large to fit in the main body of the paper.

Figure: Figure 17: Absorption rate vs L0 by letter, layers 0-17. We see a wide variance in which letters are absorbed by which SAEs.
Refer to caption: x29.png

### A.8 Feature dashboards

We include feature dashboard screenshots from Neuronpedia for some prominent latents mentioned in this work. Figure 18 shows a dashboard for Gemmascope layer 3, latent 1085, which is a token-aligned latent firing on variations of the word _short and we find absorbs the “starts with S” direction. Figure 19 shows latent 6510 from the same layer which should be the main “starts with S” latent.

Figure: Figure 18: Neuronpedia dashboard for Gemma Scope layer 3, latent 1085. This latent is a token-aligned latent for _short tokens. This latent absorbs the “starts with S” direction.
Refer to caption: extracted/5891148/figures/1085_dashboard.png

Figure: Figure 19: Neuronpedia dashboard for Gemma Scope layer 3, latent 6510. This latent should be the main “starts with S” latent.
Refer to caption: extracted/5891148/figures/6510_dashboard.png

## Peer Reviews

confidence: 3
contribution: 3
initial_rating: 8
presentation: 3
questions: Same as above
soundness: 3
strengths: 1. This paper studies an interesting problem, that is, learned sparse features can become less understandable by "absorbing" token-aligned features.
2. understanding how width and sparsity of SAEs affects its training performance can be helpful for future SAE training.
3. the paper is written in an easy-to-understand way.
summary: This paper explores the use of Sparse Autoencoders (SAEs) to decompose activations of Large Language Models (LLMs) into interpretable latent features. It addresses two main questions: the degree to which SAEs yield monosemantic (single-meaning) and interpretable latents, and how adjustments to the sparsity and size of SAEs impact interpretability. Through a controlled first-letter identification task with complete ground-truth labels, the authors find more nuanced insights compared to previous studies.
weaknesses: 1. It would be valuable to explore practical examples where insights from feature absorption can enhance interpretability. For instance, analyzing the first letters of all tokens associated with specific latent activations might offer new explanations for features that previously seemed opaque. This approach could reveal subtle patterns in latent activation, aiding in the interpretation of challenging features.
2. This work presents a specific form of feature absorption, but it raises the question of whether other variations might exist. Are there additional contexts or scenarios where feature absorption manifests differently, potentially impacting interpretability in distinct ways? Identifying these cases could deepen our understanding of the phenomenon and refine the strategies needed to address it.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 2
questions: I list my concerns and questions in the "weaknesses" section.
soundness: 3
strengths: As the large language model becomes increasingly popular, it is really important to get a good understanding of these techniques. This paper investigates a meaningful problem and attempts to provides some insights and points out some research directions, which should be very useful to the community.

#---------------------------------------#

After rebuttal, the authors add results with different SAE architectures and three different model types.
summary: This paper investigates the interpretability of the extracted latent/features by Sparse Autoencoders. The authors found that the Sparse Autoencoder does extract monosemantic and interpretable features and changing the hyperparameters of the Sparse Autoencoder could not eliminate this.
weaknesses: There are several weaknesses for this paper.
1) The observation and conclusion are merely based on one model and one task (first-letter identification task). Thus, it is really difficult to say how general and convincing these conclusions could be. I am afraid it is not safe to draw a solid conclusion based a single setting.
2)  The authors should elaborate their method better. The current version simply lists some background and then posts a pseudo code (algorithm) there. More descriptions should benefit readers to get a better understanding.
3) Some contents cause confusing instead of help. For example, the Figure1 is really confusing and hard to interpreter and even worse, the authors do not provide sufficient descriptions of it in the main paper.
4) The overall presentation is a little bit messy and it is really hard to follow the content.

---

confidence: 4
contribution: 4
initial_rating: 6
presentation: 3
questions: - Did you find any rules for predicting whether the feature activation for the first letter of a given word would be absorbed? (i.e. do you have any hypothesies/ways to differentiate situations in which feature absorbtion occurs or does not occur?)
soundness: 4
strengths: - Describes an interesting setting in which to study feature disentanglement for natural language.
- Discovers a new phenomenon, 'feature absorbtion', which could pose a problem for desired future applications of Sparse Autoencoders.
- Setting allows for a thorough analysis of the failure cases for feature absorbtion and feature splitting, and said analysis was good and informative.
- Paper has a clear statement of the problem and good structure. Good use of the 'S'/'_short' example to help build intuition.
summary: This paper proposes a new simplified setting for studying features discovered by Sparse Autoencoders (SAEs) trained on language model activations. It supposes that SAEs should discover features corresponding to the first letter of a given token, but finds that features which are predictive of a given letter occasionally fail and are 'absorbed' into the activation of another feature. They then measure and explore this phenomenon in the context of their setting.
weaknesses: - It would be useful to explore the commonalities between different instances of absorbtion of SAE features, or otherwise find more ways to verify/expand upon the claim/statement that "feature absorption is likely a logical consequence of SAE sparsity loss".

---

confidence: 4
contribution: 3
initial_rating: 6
presentation: 3
questions: **Questions** 

1. The concept of "approximately token-aligned" latents is central to claim 3 (lines 69-71), but is not defined anywhere. How is token-alignment defined, and is it quantified or measured in these experiments?

2. When the F1 score of an SAE is computed, as in Figure 2 (lines 169-172), how are SAE activations converted into a binary classification? Is the classification "does/doesn't activate". Is a non-zero threshold used? If so, how is that threshold chosen?

3. In Figure 7b, when the mean absorption rate is \~35%, is that indicating that for roughly 35% of tokens, there is a latent absorbing that token? The vocabulary size of Gemma-2B is \~256k (https://arxiv.org/html/2408.00118v1#:~:text=Vocab%20size-,256128,-256128) and 35% of that is \~90k. If that is the case, how are 90k tokens absorbed into 65k latents?

4. In figure 13, why is the high cosine similarity in the .4-.6 range, and not closer to 1? This range of cosine similarities is suggestive of learning the "true" feature mixed with a second, orthogonal feature. 

5. It seems that the paper usually uses cosine similarity of the LR direction with the encoder direction, but Figures 4 and 13 use decoders instead of encoders. Why is that?

6. What does Algorithm 1 (Lines 116-127) return? One ablation effect per token per latent? Or are these averaged in some way?

**Other Comments**

If it is permitted for authors to make revisions before the final submission, there are several small changes that could improve the quality of the paper:

- Lines 94-107: the paper should reference (Lieberum et al., 2024) since that is the exact architecture of SAE being studied. 

- Line 100: Although ReLU is often used in SAE architectures, it would be better to write a generic activation function in this equation, especially considering that the SAEs studied in the paper use JumpReLU, not ReLU.

- Line 118 and Line 147: The description "include the SAE error term" is ambiguous in the text. It is clarified in the glossary, and by the glossary's citation to (Marks et al., 2024), but the main paper neither cites (Marks et al., 2024) here, nor links to the glossary. Is there a way to include hyperlinks to the glossary in the text itself? 

- Line 205: There is a citation to (Gao et al., 2024) for sparse probing. It is likely supposed to cite (Gurnee et al., 2023) as in the background section. (Gao et al., 2024) use k-sparse SAEs, which are distinct from k-sparse probes and the SAE architecture studied in the paper.

- Lines 291-292: "The token where the SAE feature activates is highlighted in green" should be "yellow" since the highlight is yellow.

- Line 404: The paper says "it is difficult to apply [(Karvonen et al, 2024)] to existing SAEs trained on real LLM activations." One of the main contributions of (Karvonen et al, 2024) is very applicable to this paper, namely coverage (Section 3.2 in https://arxiv.org/pdf/2408.00113). Coverage for a given set of properties (in this case, "first letter identification") is defined by, for each property, finding the SAE latent and threshold which results in the best F1 score for classifying that property, then averaging those F1 scores across the properties. This is very similar to what is shown in Figure 2, though with a different selection rule for SAE latents. The paper could make reference to the similarity of this method to coverage.

- Line 429, possible typo: "..find that by training an SAE on the decoder *or* another SEE, a technique...", the word "or" should be "of".

- (Uncertain) The citation of "(Huben et al., 2024)" should instead refer to the paper as "(Cunningham et al., 2024)". The order of the authors is different between the OpenReview page (https://openreview.net/forum?id=F76bwRSLeK) and the paper itself (https://openreview.net/pdf?id=F76bwRSLeK). In such a case it is most likely proper to follow the order of authors in the paper.

**References**

Leo Gao, Tom Dupre ́ la Tour, Henk Tillman, Gabriel Goh, Rajan Troll, Alec Radford, Ilya Sutskever, Jan Leike, and Jeffrey Wu. Scaling and evaluating sparse autoencoders. arXiv preprint arXiv:2406.04093, 2024.

Adam Karvonen, Benjamin Wright, Can Rager, Rico Angell, Jannik Brinkmann, Logan Riggs Smith, Claudio Mayrink Verdun, David Bau, and Samuel Marks. Measuring progress in dic- tionary learning for language model interpretability with board game models. In ICML 2024 Workshop on Mechanistic Interpretability, 2024.

Wes Gurnee, Neel Nanda, Matthew Pauly, Katherine Harvey, Dmitrii Troitskii, and Dimitris Bertsi- mas. Finding neurons in a haystack: Case studies with sparse probing. Transactions on Machine Learning Research, 2023. ISSN 2835-8856. URL https://openreview.net/forum? id=JYs1R9IMJr.

Robert Huben, Hoagy Cunningham, Logan Riggs Smith, Aidan Ewart, and Lee Sharkey. Sparse autoencoders find highly interpretable features in language models. In The Twelfth International Conference on Learning Representations, 2024. URL https://openreview.net/forum? id=F76bwRSLeK.

Tom Lieberum, Senthooran Rajamanoharan, Arthur Conmy, Lewis Smith, Nicolas Sonnerat, Vikrant Varma, Ja ́nos Krama ́r, Anca Dragan, Rohin Shah, and Neel Nanda. Gemma Scope: Open Sparse Autoencoders Everywhere All At Once on Gemma 2, August 2024.

Samuel Marks, Can Rager, Eric J. Michaud, Yonatan Belinkov, David Bau, and Aaron Mueller. Sparse feature circuits: Discovering and editing interpretable causal graphs in language mod- els. Computing Research Repository, arXiv:2403.19647, 2024. URL https://arxiv.org/ abs/2403.19647.
soundness: 3
strengths: The paper explores a natural task, first letter classification, and does so well. It establishes that this information is linearly available in the model (by training an LR probe with high F1-score), and that SAEs find latents that align with the probe direction (Figure 4b and especially Figure 13), and that these latents are often good classifiers, though not as good as probes (Figure 2a).

The paper provides a novel perspective to a documented issue of feature splitting in SAEs. It provides good theoretical justification of why feature absorption is likely to happen, and why it provides challenges to interpretability.
summary: This paper documents and measures a particular form of feature splitting in the Gemma Scope family of sparse autoencoders (SAEs), which the authors call feature absorption. The prototypical case of feature absorption studied is when a latent is a high-precision classifier for a property like "starts with the letter L", but has poor recall due to other latents "absorbing" individual tokens, so that the main latent's activations are given by the rule "starts with the letter L, except for the tokens laser, lions, [several other exceptions]".
weaknesses: The paper could do more to demonstrate that feature absorption happens as described, namely that: 
1) There is a latent which is a high-precision classifier for "starts with letter [x]".
2) There is a particular subset of tokens which start with this letter but which the latent fails to activate on.
3) There are separate token-aligned latents that individually classify those tokens.
4) If that subset of tokens is removed, the latent is a high-precision, high-recall classifier for "starts with the letter [x], except for that subset of tokens".

The authors demonstrate 1) and 2) well in sections 4.1 and 4.3 respectively, but never show 3) or 4) besides a single example in section 4.2. The authors could address 3) by adding an additional step to the experiment in section 4.3, in which they confirm that "the SAE latent receiving the largest negative magnitude ablation effect" is a token-classifying latent.

Lines 148-157: The metric described can fail at its stated goal to "[measure] the propensity of the model to choose the correct starting letter as opposed to other letters" because averaging logits can interact poorly with the softmax activation function. For example, if "A" is the correct letter, but the model logits are g[A]=10, g[B]=25, g[C]=g[D]=...=g[Z]=0, then the model would produce B with near-certainty. But the provided metric yields m=10-(25+0+0+...+0)/25=10-1=9, the same value as if g[A]=9, g[B]=...=g[Z]=0. This metric would be more convincing if the average were replaced by a max, or if it were a softmax over letter tokens. 

The authors only study a single family of SAEs on a single language model. Alternative SAE architectures may not demonstrate feature absorption.

Meta-review: Sparse Autoencoders (SAEs) have become a popular tool for interpreting neuron activation behaviors in large language models (LLMs). This paper questions the effectiveness of sparsity regularization in SAEs by identifying the phenomenon of feature absorption—a behavior where overlapping features are absorbed into a single latent representation, degrading interpretability. The authors use a controlled first-letter identification task to analyze feature absorption, demonstrating its prevalence and persistence across varying SAE sizes and sparsity levels. They identify sparsity penalties in SAEs as the root cause of this phenomenon.

Multiple reviewers appreciated the identification and analysis of feature absorption, describing it as a meaningful contribution to the understanding of SAEs. The use of controlled settings and diverse models (e.g., Gemma Scope, Qwen2, Llama) provides robust empirical validation.

Concerns from the reviewers:

- Limited Real-World Applicability: The study focuses primarily on the first-letter identification task, limiting its generalizability to broader real-world tasks. 

- Unexplored Variants of Absorption: Reviewer S6kU and others noted that additional manifestations of feature absorption or scenarios where absorption may behave differently were not explored, leaving gaps in understanding.

- Presentation Issues: Multiple reviewers criticized the paper’s overall clarity and organization, pointing out ambiguous descriptions and confusing visuals (e.g., Figure 1). 

**Recommendation**

While there is considerable interest in the identification of feature absorption, as reflected in reviewer and public comments, and all reviewers recommended acceptance, the paper’s shortcomings are substantial:

1) Its focus on a toy problem raises question on the generalizability and practical relevance of its findings.

2) there is no rigorous theoretical characterization of the phenomenon, limiting the depth of its contributions.

3) it has limited attempts at addressing the identified problem. For example, if the issue is with hierarchical features then methods from hierarchical sparse regularization (see e.g. [a]) would be a natural alternative to (unstructured) sparse coding?

4) the lack of clarity in the presentation raises the concern that the paper may not be broadly accessible to the broad audience of ICLR. In particular, reviewer 1gVz raised multiple concerns on clarity of the presentation but there was little effort during the discussion phase to address those.

Given all these, I’d recommend a weak rejection while encouraging the authors to improve their manuscript along the lines above to substantiate their interesting findings.
 
[a] Proximal Methods for Hierarchical Sparse Coding, JMLR 2011

Decision: Reject