Sequences of Logits Reveal the Low Rank Structure of Language Models
Authors: Noah Golowich
Allen Liu
Abhishek Shetty
Venue: ICLR 2026 Oral (2026)
Keywords: Large language models
low-rank structure
Primary area: learning theory
TL;DR: We exploit the low-rank structure of the logit matrices of LLMs to draw new empirical and theoretical conclusions.

Abstract: A major problem in the study of large language models  is to understand their inherent low-dimensional structure.  We introduce an approach to study the low-dimensional structure of language models at a model-agnostic level: as sequential probabilistic models. We first empirically demonstrate that a wide range of modern language models exhibit low-rank structure: in particular, matrices built from the model's logits for varying sets of prompts and responses have low approximate rank. We then show that this low-rank structure can be leveraged for generation --- in particular, we can generate a response to a target prompt using a linear combination of the model's outputs on unrelated, or even nonsensical prompts.

On the theoretical front, we observe that studying the approximate rank of language models in the sense discussed above yields a simple universal abstraction whose theoretical predictions parallel our experiments. We then analyze the representation power of the abstraction and give provable learning guarantees.

Published conference paper at ICLR 2026
SEQUENCES OF LOGITS REVEAL THE LOW RANK STRUCTURE OF LANGUAGE MODELS
Noah Golowich
Allen Liu"
Abhishek Shetty* MIT
Microsoft Research
UC Berkeley a 44 edu
nzgêcs utexas.edu
shet.tymitt.es
ABSTRACT
A major problem in the study of large language models is to understand their inherent low-dimensional structure. We introduce approach to study the low- dimensional structure of language models at model-agnostic level: as sequential probabilistic models. We first empirically demonstrate that a wide range of mod- ern language models exhibit low-rank structure: in particular, matrices built from the model's 's logits for varying sets of prompts and responses have low approximate rank. We then show that this low-rank structure can be leveraged for generation in particular, we can generate a response to target prompt using a linear com- bination of the model's 's outputs unrelated, or even nonsensical prompts.
On the theoretical front, we observe that studying the approximate rank of lan- guage models in the sense discussed above yields simple universal abstraction whose theoretical predictions parallel our experiments. We then analyze the rep- resentation power of the abstraction and give provable learning guarantees.
INTRODUCTION
Understanding the structure of language has been a long-standing goal in computer science, motivat- ing various fundamental models (Shannon, 1951; Chomsky, 2002) such as Hidden Markov Models (HMMs), finite state automata, and formal grammars (see (Jurafsky & Martin, 2025)).
Such questions have taken on new light in recent years with the advent of Large Language Models (LLMs); despite extensive study, though, their success at modeling language has largely eluded mathematically rigorous understanding. Such an understanding, while of interest in own right, is also crucial when it comes to evaluating the possible risks associated with LLMs. Indeed, sizeable literature has uncovered surprising ways to bypass the safety mechanisms of LLMs (e.g.. (Wei et al., 2023; Zou et al., 2023)) as well ways they can become misaligned during training (Razin et al., 2024; Betley et al., 2025). Understanding and defending these vulnerabilities often requires mathematical insight into the model's workings (Elhage et al., 2021: Olsson et al., 2022; Turner et al., 2023; Arditi et al., 2024).
One of the principal ingredients still lack when it comes to obtaining more solid theoretical foundation for studying LLMS are simple universal abstractions that are realistic enough to make testable predictions about deployed LMs and mathematically tractable enough support prov- able guarantees. Many such models have been proposed, ranging from simple types of transformers (Sanford et al., 2024), to low-depth circuits (Merrill & Sabharwal, 2023). While such frameworks have led to important insights, e.g., regarding expressivity. most of them are limited in the types of models they can represent (c.g., transformers of a certain depth (Merrill Sabharwal, 2025)) or to certain types of tasks (e.g., RL. fine-tuning (Foster et al., 2025)). Further, they are often not very capable of making precise predictions about the structure and behavior of modern LLMss.
In this paper, inspired by the folklore belief that language possesses intrinsic low-dimensional struc- ture, we propose an crchitectuur-agnnssiie framework which allows us to study how information in language is represented in low-dimensional subspaces. In particular, we study logit matrices associ- ated to any language model (Section 1.1). This approach treats the language model as a sequential
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(a) OLMo-7b low-rank approximation
(b) Performance of LINGEN with OLMo-1b
Figure 1: (a): Low-rank approximation error (measured by average KL divergence; see Theo- rem 3.1) of the extended logit matrix for OLMo-7b, and ranks 5-500. For fixed sets H,F, the approximation errors for the logit matrix C (H, behave according to a similar power law as to those of various sub-matrices with (2, 4, 8, 1-)-temss fewer entries. Dashed line top shows perfor- mance of a (suboptimal) rank-1 baseline. (b): Performance of our generation procedure (LINGEN; star markers) which exploits the low-rank structure of the extended logit matrix to generate from a given "target" prompt by only querying the language model on nonsensical prompts unrelated to the target. We plot the KL divergence between LINGEN and the true language model (OLMo-I) 1b) at each token position, as well as various baselines (solid lines; see Section 3.3).
probabilistic map from sequences to sequences, sidestepping architecture-specffic details. start- ing point for our work is the observation that the logit matrices associated to modern autoregressive LLMs are wepl-appro by low-rank matrices (Section 3.1). Additionally, this low-rank struc- ture can be exploited to exhibit several surprising phenomena, such the ability to generate from a given model by only querying it on unrelated or even nonsensical prompts (Sections 3.2 and 3.3). Moreover, such low-rank assumption yields rich theoretical landscape, from both learning- theoretic and generative modeling standpoint (Section 4).
1.1 STRUCTURE ARISES FROM THE EXTENDED LOGIT MATRIX
Our goal to study low-dimensional representations of language models: what aspects of the model should such a representation capture? One natural requirement is that it allows us to compress the past in order to generate the future. Perhaps the most fundamental and extensively type of such compressions tokens h (e.g., a prompt), the compression consists of a vector Φ(h) E Rd so that the probability of observing any possible completion f (e.g., response) is proportional some embedding vector (f) E TR' depending only on f.
those which are linear, in the following sense. For a sequence of
(f))), for
We remark that very similar setup has been considered in the context of LLMs, when the sequence I consists of single token: for essentially all modern LLMs, due to the structure of the unembed- ding matrix, the probability of observing a single token y following sequence h is proportional to exp( (o(h), (z))), for some embedding maps o, sb This property has been termed the softmax bottleneck (Yang et al., 2017). and has had a broad array of implications, ranging from expressivity and architecture (Press & Wolf, 2016; Kanai et al., 2018; Ganea et al., 2019; Chang & McCallum, 2022; Godey al., 2024) to security (Carlini al., 2024; Finlayson et al., 2024). However, this observation only allows to reason about one token in the future. The key insight of our work is that this low-dimensionality persists, even when we consider logits over longer sequences of tokens.
To study this low-dimensionality over longer sequences, introduce the extended logit matrix as our main object of study. Its rows are indexed by all possible sequences tokens, which we refer to histories. Roughly speaking, its columns history; we refer to them as futures. In essence, the entry of the extended logit matrix corresponding to given (h, f) pair and a language model M is given by log Pr [f /t). i.e., the logarithm the probability that M assigns to f given context h. This
also indexed by all possible sequences, to be
interpreted as possible completions
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choice is natural given the above discussion- indeed, the fact that log Pr h)] ~ (Φ(h), +())
for some feature mappings Φ, y (as discussed above), is equivalent to the following statement:
The (extended) logit matrix is approximately low-rank for modern language models.
The main premise of this work is that the above statement is true for modern autoregressive LLMs. While we cannot construct the entire extended logit matrix (which is exponentially large), we can construct submatrices indexed by sets of histories H. and futures F. and measure how the low-rank structure of these submatrices (denoted Cas (H. evolves with their size (Figure 1). As is shown, this approximate low-rank structure is remarkably consistent as we scale up the number of histories and futures. We emphasize both that this low-rank structure is atypical for (random) matrix of the same dimensions and that it is not a consequence of the low-rank structure of the single-token logit matrix (see Section C). For the formal definition of the extended logit matrix, which differs slightly from the simplified description here, we refer the reader Section 2.
1.2 OUR CONTRIBUTIONS
In this paper, we conduct a thorough empirical study of the extended logit matrix across both a wide range of LLMs and choices for the histories and futures, and complement our empirical findings with theoretical results on expressivity and learnaillity. Our framework suggests a number of intriguing open questions relating to interpretability, safety, and the theoretical underpinnings of L.MMs.
Empirical Findings:
low-rank structure. In Section 3.1, we study the extended logit matrices defined by a wide range of language models, and sources of histories and futures. We observe that the extended logit matrices are all well-approximated by low-rank matrices, and that the quality of the best low-rank approximation follows power law which is strikingly consistent as the matrix is scaled up. We also observe that the low-rank structure not present at the beginning of training, but rather emerges in the early stages of pre-training and further evolves throughout training.
Empirical Findings: consequences of low-rank structure. Given that the extended logit matrix is typically well-approximmted by low-rank matrix, natural follow-up question is to understand the implied linear dependencies between the rows (i.e., histories) of the matrix. Such linear depen- dences may be seen as generalizations of classical examples such as boy girl as king queen (Mikolov et al., 2013). In Section 3.2, we verify that these linear dependencies are consistent across the extended logit matrices of different LLMs. More surprisingly, they remain preserved even if we significantly change the futures (i.c., columns) of the matrix, replacing the original futures with nonsensical futures consisting of random sequences of tokens.
The fact that an LLM's low-rank structure is preserved even when prompted with nonsencial (ran- dom) sequences inspires us to investigate (in Section 3.3) the following setup for language gen- eration. Given any target prompt (i.e., history), using an appropriate extended logit matrix we can approximately write as a linear combination of unrelated, even nonsensical, histories. We can then use this linear combination to generate a continuation to the target, by only querying the model on the unrelated histories. We find that this procedure generates coherent continuations, and its KL-divergence to the true model beats strong basclines, c.g., intermediate training checkpoints (Figure lb). Such a procedure has the potential to circumvent defenses, such as prompt filters (Dong et al., established to ensure safety; we discuss further implications for Al safety in Section 3.3.
Theoretical Framework. In addition, a key aspect of our framework is that it supports a simple theoretical generative model that captures the low-rank structure that we find. Our empirical ob- servations inspire us to revisit the Input Switched Affine Network (ISAN) of Foerster et al. (2017). ISANs were initially proposed as a simple, interpretable recurrent architecture, where a low dimen- sion hidden state is updated through linear transformation that is allowed to depend on the current sampled token, and were shown to achieve reasonable performance on language modeling tasks. We show that ISANS capture precisely those distributions that have (exact) low logit rank (Theorem 4.3).
To support the tractability and usefulness of ISAN as a theoretical model, we first study its repre- sentation power, showing that it can express state space layers, the key component of a variety of practically successful architectures broadly termed state space models (SSMs (Gu et al., 2021a; Gu & Dao, 2023; Gu, 2025)). We also show that it can express algorithmic behaviors like copying and noisy parity - the latter having implications for learnability. Since noisy parity is hard to learn
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(Blum et al., 2003), efficiently learning an ISAN from samples is impossible in the worst case. In light of this, we give provably efficient learning algorithm with lagit queries, a setting that closely resembles practical model stealing setups for common APIs (Carlini et al., 2024).
Related work.
We give a more detailed discussion of related work in Section A.
2
SETUP AND NOTATION
Our main object of study is autoregressive language models. We denote such models with the letter M and the associated set of tokens with Σ. For a token 2 E Σ and a context sequence Viet = UVl,.....1)) E Et, we will denote the corresponding probability distribution over the next token by PrM(- yı:)), i.e., Prw[a [= yia] is the probability that the next token is E Σ. We let E and بو represent sequences of length f and at most t, respectively, while D denotes the set of all sequences of tokens. Note that these include the empty string, which denote by Null. We use to denote the concatenation of two sequences, i.c., y o y denotes the concatenation
and y'.
To formally introduce the key object that we study, the extended logit matrix, first define the mean-centered logits; while empirically mean-centering doesn't i make a large difference, will be important for the theoretical generative model that we introduce later (see Remark 1).
Definition 2.1 (Mean-Centered Logits). Given a sequence Y1st E Σ', we define the mean-centered logits as Lu [z|ya=]] log Pr | [=|vaa] [s]] Eves log Pr [='[vaa]
Σ.
Given model M and sets H.F Σ" consisting of sequences of tokens, the extended logit matrix (H, has rows indexed by H. and columns indexed by F x E; accordingly, we will write CM, F) E IR (xFxxx) We refer to elements of as histories and of F as futures. Formally: Definition 2.2 (Extended Logit Matrix). Fix model M. Given subsets of histories H C Ex and futures F C E", the associated extended logit matrix L (H, (Hx((xE)) is defined for h e H. and (f, 2) F by
LM, F))n.)))) LM[= J.
Schematically, for histories H (7r -....... hn) and futures F (fi,-..,fr fF], the matrix LM (H,F) F) has one block of length |>]] for each history-future pair (hi, fu) and that block contains the (mean-centered) next-token logits for fz:
{Lar(> h1 o fillser
{Laslz h1 o frllze
F),
: Laal>> hn frllee
[LM[ hu fillee
In the remainder of this paper, we will refer to the extended logit matrix as simply the logit matrix.
Relation Simplified Definition in Section 1.1. The definition above, while slightly modified from the definition introduced in Section 1.1, serves as linear representation in almost the same sense. Observe that we set F EST then a row La ([h). contains all of the information necessary to sample a continuation of h of up to length T. This is because, from the information in this row, we can compute log Pr[f[ for any f ΣST by breaking up f 21:t into tokens and summing up the token-by-token log conditional probabilities
log [r[ = log Pr(z. [h 21:-1] +... log Pr[21 [h]
If we ignore the normalizing constant and replace the mean-centeree logits with normalized logits, then the above is actually linear function of the entries in the row ((r(h),, The main purpose
of writing the extended logit matrix in this new form in Theorem 2.2 is that the logits can be more
directly extracted from the model.
3
EXPERIMENTS
3.1 Low RANK STRUCTURE
In this section, we evaluate the extent which the logit matrices associated to modern LLMMs (The- orem 2.2) are actually low-rank, To do so, we computed logit matrices Las (H, corresponding

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to various models M and sets H.F of sequences. The sets H.F were generated as follows starting from a dataset D, according to some parameter n E N. We sampled 2n sequences from D and for each of the 2n sequences included random subsequence of it in either F or H. (so that each of F, H. ends up with n subsequences). We studied a broad range of datasets D and open-source models M: precise details regarding the choices of D, M as well as H.F may be found in Section B.1.1. For the figures shown in the main body of the paper, the dataset D was always taken to be the wiki split of o 11mo-mix- mix-mixxxie 1224 (OLMo et al., 2024).
We considered various values of n, ranging up to 10a. Note that the full logits matrix CM (H, F) has n² |Σ] entries, which is not feasible to store given that for many models (e.g., OLMo) |Σ| R 10". Thus, in our experiments we used instead a sub-matrix of Lss FH, which we denote by L, (H, F), for some parameter k. The submatrix (H, F) is obtained by selecting subset of the columns as follows: for each future f E F, we select the columns indexed by (f, z) where z is one of the A: likely tokens following f. ie.. for which Pr [z f] is largest. In our experiments we took k = 50; in Section 1.1.5 we also repeated some experiments for (a) 1: = 200 and (b) where a random subset of k 50 tokens was selected for each f. and observed similar results. At a high level, LM. LM,*(H,,)) can be interpreted as containing information about the model's predictions for the most-likely next tokens for each possible context h f (with E H. E F). We measured the degree to which the resulting logit matrix (H,, is well-approximated by low-rank matrix in two ways, discussed below. For ease of notation, we omit the parameter & (and ignore the distinction between a (H, F) and (H, F)) in our description of the experiments throughout this section.
Method I: measuring the singular value decay. In Fig- ure 2 (and Figure 7, which shows additional LLMs), we display the singular values of the logits matrix (H,,H, F) for various choices of models M and various sizes of the sets H, when the dataset D was wiki. The singular val- ues decay approximately according
by
Homaired S Sngular Salue
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power law, in that the ith singular value o; of the matrix (H, FF) is approx- imately C P.ii- i- for some C, > (depending on M).
As shown in Figures 2 and 7. the exponent a for most logit matrices is slightly greater than 1 1/2: this holds as well for additional choices of the dataset D (see Figure 9 in the ap- pendix). Intriguingly. a = 1/2 represents a phase tran- sition for low-rank approximation, in the following sense (see Fact B.1 for a formal statement): a > 1/2, then for any constant c, there is a constant T= depending only on E so that rank-r, matrix e-approximates the logit matrix. On the other hand, if a < 1/2, then for sufficiently small constants E, a rank linear in the dimension is needed.
Singuler Value Index
Figure 2:
OLMo-7b singular values; Power law exponent a Rs 0.536
When it comes to approximating the logit matrices of language models, are most interested in the setting where H.F contain all possible histories and futures (or at least those that are plausible in natural language), information in the language model. In this setting, the logit matrix's 's dimensions are exponential in the length of the sequences, meaning that the case of power law decay with a < 1/2 requires that the rank be exponentially large to achieve approximation error less than some absolute constant. This is in contrast to constant rank for the case > 1/2 which we observe in Figures 2, 7 and
then the logits matrix Ls FH, contains all the next-token prediction
While it is infeasible to compute the logits matrix when H.F are taken to be the sets of all histories and futures,
apparent the figures that the exponent of the power law remains essentially unchanged when we consider sub-matrices of LM(HH, (modulo some degeneration of the power law at ranks that approach the size of H. F, which is expected). Thus, as H. F are scaled up, we expect that the same power law holds.
Method 2: measuring the approximation via KL divergence. While measuring the decay of
singular values of matrix is mathematically convenient (in the sense of, e.g., Fact B.1), it does not directly yield a probabilistic interpretation of the fact that the logit matrix (HH is close to another (low-rank) matrix. To bridge this gap, we observe that any matrix L E (F (c.g.. a logit matrix Las (H, induces a collection of [ "next token distributions" on Σ by taking the softmax of the vector Ln.f == (I) for each h f. (Indeed, recall from Theorem 2.2
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that when L - FH, we have La.,- Lu f.. In place of Frobenius norm, we propose to measure the average KI. divergence between these distributions:
Definition 3.1 (Average KL divergence). Fix sets H,F -", and consider matrices L, A (xx(FxE), We define DLL (L, A) to be the average KL divergence between the next-token dis- Iributions induced by the entries of Ln.s and An, for each pair::
pair:
DRL (L, A)
F
REN.SET
Thus, for low-rank matrix A, the quantity DKL (Car F), may be interpreted as the ability of the low-rank matrix A to approximate the model M in KL divergence, when restricted to contexts o for H f F. It straightforward show (see Fact B.2) that we may bound
(H, S ममोन (H,, - All Combining this fact, Fact B.I, and the ob- served decay in singular values in Figure 2, we should expect the average KL divergence between CM, F) and a rank-r approximation to decay at least as fast as power law (in r). This predic- tion is confirmed in Figure la (see also Figures 8 and 10 in the appendix), where also observe consistency in the power law as the matrix size is increased, suggesting that such a low-rank ap- exponentially large.
proximation holds when H. F are
Evolution of approximation during the course of train- ing. One intriguing exception the findings discussed above is that for a "baseline" model M corresponding to the "Step O" checkpoint of OLMo- lb (i.e., before any training steps), the logit matrix does appear to exhibit low-rank structure: see Figure 7d in the appendix, where we have 0.374. Further, while the KI. divergence approximation does decrease with the rank (Figure 8d), this decrease does not seem follow consistent power law as the matrix's 's size is increased, suggesting that for larger sets H, F such a decrease may be diminished. These observa- tions lead us to ask: At what point in training does the logit matrix become (approximately) low rank?
KL Divergenoe
low-rank
Rank
Figure 3: Low-rank approximations (wrt. avg. KI. divergence) over Stage- pretraining of OLMo-1b.
To answer this question, we show, in Figure 3, the average KL divergence to low-rank approxi- mation for the logit matrices at various pre-training checkpoints of OL.o 1b (given a fixed set of H.F each of size approximately 4000). Early in training, the KL divergence to low-rank approxi- mation drops significantly, then rises slowly towards its final value. This observation suggests many fascinating questions for follow-up work, foremost amongst them: How and why does low-rank structure in the logit matrix emerge early in pre-training?
3.2 SHARED STRUCTURE ACROSS MODELS & FUTURES
Next, we proceed to discuss some consequences the (approximate) low-rank structure of the logit matrix Lxr (H, One of the most basic facts about low-rank matrices is that they have nontrivial (row) kernels, i.e., if a matrix A E (FxxE) (say, an approximation of LAs (H. F)) has rank much less than [71), then there is a large space of nonzero vectors v E HR for which
What does the existence of such vectors mean? Roughly speaking, such vectors e represent "linear relationships" between different histories. Studying such linear relationships has been a mainstay of NLP over the last decade, with simple example being relationships between word embeddings such as: boy - -irl Re king- - queen (Mikolov et al., 2013). When comes to logit matrices, do such linear relationships manifest in ways that correspond to semantic relations between histories? Towards answering this question, we first describe the following "sanity check": for the sets of histories H and futures F considered in Section 3.1, for each h E H computed the h' + h in H minimizing the norm of the difference between the corresponding rows of the logit matrix, namely LM (1h)), Lar (fh'), Some samples of resulting pairs are shown in Table where it can be seen that most of the pairs of histories share semantic similarities.

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Kask: 10
Systematic evaluation: description. We aim to more systematically evaluate whether vectors E R# describ- ing linear relationships between rows (i.e., histories) of A R LMr(H F) are inherent to the histories themselves, as opposed to being spurious artifacts of either the model architecture or the set of futures F. If indeed such vec- tors were not spurious, then would expect that: the induced linear relationship given by such vectors transfers (a) across the choice of futures F and (b) across models M. In other words (regarding (a)): given distinct sets of futures F,F' (drawn from different distributions), after choosing low-rank matrices A, A' satisfying A ~ Lu FH, and A' = Cas (H,, the row kernels of A, A' should be roughly equal. This would mean that approx- imate linear relationships between the rows of Lm (H, F) are also approximate linear relationships between the rows of Lms (H, F'), and thus these relations are between the histories themselves, independent of the choice of futures.
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Figure 4: Cos of principal angles be- tween column spaces of low-rank apx of L (H.F) & (M,H, ).
Equivalently, since the column span of a matrix is the orthogonal complement of the matrix's row kernel, we expect that the column spans of A, A' should be roughly equal. Moreover (regarding (b)), a similar conclusion should hold if A, A' were low-rank approximations of LM,H, and LM: (M.(H, FF. for distinct models M, M.M
Invariance the choice of F: "nonsense" futures. To evaluate the above hypothesis regarding differing sets F, of futures: for fixed model M (taken to be OLMo-Ib), and sets H.F as con- structed in Section 3.1, we create a "nonsense" version J.nossense of J. by randomly permuting all tokens amongst futures of F. Given a parameter E: N, we let A E (F (resp., Anonsensee de- note the best rank-r approximation to (HL,F) (resp., the column spaces of A, Anonsense approximately equal? In Figure 4, we answer this by plotting, for various values of r, the (cosines of the) principal angles between the column spans of A, A nonsense A large fraction of the principal angles are close to 0, i.e. have cosines close to (and this fraction is much larger than for pair of independent uniformly random r-dimensional subspaces, which is shown in the dashed lines). This means that the column spaces of A, A nonsense are mostly overlap- ping. and thus linear relationships between histories transfer to significantly different sets of futures.
Fhonsesse) in Frobenius norm. Are
The above experiment demonstrates that the model's logits Las [- ho fnonsense] ), for history h E H and a "nonsense" future fronsense E Fnhonsense, still yield significant information about h o f. for real futures f F. In partieular, given vector representing some linear relationship between histories in the sense that o. (H,, F.nosense) Rs O, we typically have also v LL LH, F) R O.
Invariance to the choice of M. In Figure 12 in the apppendix, we test the additional hypothesis of whether the linear relationships between histories are shared across different models. We observe similar overlap between the column spaces of low-rank approximations to (H, for various pairs (M, M') of distinct models. This means that, at least to some extent, these linear relationships are inherent to the sentences themselves, independent of the choice of model.
3.3 EXPLOITING Low RANK FOR GENERATION
Finally, we investigate one consequence of the fact that there are "linear relationships" between histories as captured by vectors E H satisfying uT_ Las La, (H,F) z O. and moreover that these relationships transfer to (potentially unrelated) sets of futures. We will show how to exploit these relationships to approximately generate samples from a language model M conditioned on some "target history", by only querying M on unrelated sequences. In particular, consider some sequence harg E D"; typically, to generate continuations f (Z1...., Em) of hearg under M, having gener-
"Given two d- d-dimensional subspaces S' R., the principal angles between S, S' are collection of d real numbers 0,, ..,..., Au [0, x /2] with 0 < < t. which represent how much S, S' overlap: if S' then we have 0, = 0 for all i, and if S, S' are orthogonal then we have 0, = x/2 for all See (Golub & Van Loan, 2013, Section 12.4.3) for further discussion.
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212-1
h2
Zpz-1
Query, fof next- token distrs.
hg
Linearly combine using v & then sample
Figure 5: Linear generation (LINGEN) procedure for generating text from a target history using linear combinations of other histories H ha,
ated R1s--1, we query PrMv-1 hearg 21:-1] and sample on from it. (For & 1, the sequence 21.0 is to be interpreted as the empty sequence.)
We ask: Can we generate continuation of Mtarg above without making any queries of the form PrM[-1 htrg o ?1.t-1]? To answer this question, suppose we are given a set of histories H. C Σ", as well as a vector u = (v) hew R2 One should interpret e as encoding a way to write harg as a linear combination of h H in the sense that for a generic set F of futures we have LM (Mtarg: F) Re vT LH,, If this holds, then for "typical" sequences =1:-1 we have generated, we can hope that M's next-token distribution, LAs ((hearg): {z1=-1)), is close to v. L (H, {21a-1H).
The above intuition suggests the following procedure to generate a sequence of tokens 21, ....... For each 2 1, having generated 21:t-1, for each h E H. compute the next-token logit vector Ln,s == Lasl- h o 21--11 E R corresponding to h. Then we sample the next token 2, from the linear combination of the vectors Ln,t as induced by v. i.e., 2n ~ softmax (Enexy oh Ln.t). and repeat for some number m of steps. We call this procedure LIN- GEN; see Figure 5 for an overview, Algorithm (in the ap- pendix) for a complete description, and Section D.4 for a proof that LINGEN generates approximately from M's dis- tribution, under appropriate assumptions paralleling the in- tuitions discussed above. We instantiate LINGEN in two different manners:
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24
Figure 6: LINGEN with OLMo- -b.
Option 1: In-distribution target. In our first experiment, the target herg is "in-distribution" for the set of histories H we use in LINGEN: in particular, the H is taken be subsequences of wiki sequences and hearg is taken to be a subsequence of a wiki sequence distinct from all of those producing H.. We considered 50 different choices of hearg and for each generated 5 sequences of length 15 according to LINGEN. The results are shown in Figure 6: we display, at each token index I, the KI. divergence between the distribution of the next token according to LINGEN (namely, softmax (E 250 generations. The "total" numbers reported in the figure represent the average KL divergence, summed over all 15 positions. See Table 2 for some samples generated from LINGEN.
Inn))) and the model M (namely, Pra [- | h =1;t-1)), averaged over all
To compute the coefficients v for use by LINGEN, we regressed the rows of Las onto LM (harg F). Doing requires knowledge of La (htagg F) and thus requires Pr Marg o fl for each ff E F. We expect that, however, by using our observations around model transferability from Section 3.2, such vector can be computed instead using some other (perhaps less powerful) model M'. We leave further investigation on this point to future work.
query
In Figure 6, also display several baselines in which LINGEN is replaced either by (a) in orange, a version of LINGEN where u is only computed by only using, for each history, the next-token

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logits corresponding to the empty future (i.e., Ls (H, (0))), (b) in green, a version of M with restricted context window; or (c) red, an earlier training checkpoint of M. We emphasize that (a) performs well at generating the first token 21 (as expected), but performs significantly worse at later tokens Et, t > 1: this point emphasizes the importance of considering the extended logit matrix LM (H, FF as opposed to the single-token logit matrix which has been considered in prior works (Yang et al., 2017). Altogether, LINGEN performs significantly better than all baselines." 2
Option 2: Out-of-distribution target (& futures). Next, we consider a more challenging setting where the target harg is unrelated to the set of histories H. used in LINGEN. We took the same 50 targets hearg (origining from wiki) as described above, but now let the set of histories H. be the set Hnonnsnse obtained by randomly permuting all tokens amongst all elements of H. Moreover, when computing v. we replace the set of futures F with Jnnonsensee (i.e., obtained by randomly permuting all tokens). The results are shown in Figure lb; while the KI. divergence corresponding to LINGEN are generally larger than in Figure 6, they are still smaller than all baselines, and the generated sequences (Table 3) generally show clear evidence of incorporating information from A Thus, by using the low-rank structure of the logit matrix, we can generate starting from a prompt by only querying the model at (nonsense) sequences unrelated to the prompt. While we leave more thorough investigation for future work, we believe that this approach suggests novel ways create jailbreaks for LLMs, an area that has seen extensive work recently (Wei et al., 2023; Yi et al., 2024): for instance, querying the model on sequences in Hmoosense as above may allow us to eircumvent input filters which aim to filter out harmful prompts; see Section E for further discussion. In addition to implications for AI safety, we stress that the structure we observe is fundamental and could lead to many other applications ranging from computational efficiency at inference time to improved training procedures. We discuss this and other future directions in Section E.
THEORETICAL RESULTS
In the following sections, we will develop theoretical foundations for understanding low logit rank. Definition 4.1 (Logit Rank). A language model M has logit rank d for any set of histories H Σ" F C Σ", the matrix (H, has rank at most
We will show that low logit rank is equivalent to a simple generative model and then explore the rep- resentation power of this model. Finally, we will show that given query access language model with low logit rank, we can efficiently learn a description of a language model that approximates well in TV distance. These results taken together establish low logit rank natural and tractable theoretical model for understanding language models.
4.1 A Low RANK GENERATIVE MODEL
Our first main result demonstrating that the low logit rank condition can be captured by a simple generative model. The idea is to consider version of a linear dynamical system model but with two important differences: (1) the dynamics are allowed to depend on the current sampled token and the current timestep and (2) the observations generated through a softmax nonlinearity. This model is an extension of the Input Switched Affine Network (ISAN) model studied by Foerster et al. (2017) an interpretable recurrent architecture (see Theorem D.1), with the difference being that our model allows the dynamics to change every timestep. We call our model a time-varying ISAN.
Definition 4.2 (Time-varying ISAN). A time-varying ISAN of sequence length T is specified by matrices Aet E Rddd B, E Ixd for E E, E [T' and an initial state xo Rd It defines a distribution over sequences of tokens 21, 22, 2T by: 2; is sampled from softmax( B,x+-1). Then, the hidden state is updated as I, 12,, We refer d as the hidden dimension".
II general, LINGEN improves at later tokens; we believe this holds since for such tokens, the distribution of the next token depends more on the "more recent" previously generated tokens 21:-1 (as opposed to htag),,
and these tokens are fed into the model to compute Lh.t. One exception is that LINGEN performs very well the first token, which may be since the single-token logit matrix LLMs is low-rank (see Section l).
"Note that incorporating a bias term, i.c. 2, A + b...c. c. is equivalent to adding an extra dimension that is always 1. Thus, we will use the above form without the bias term.
'For simplicity, we will not have an end-of sequence token, but will treat the time-varying ISAN language
model as defining distribution over sequences of a fixed length.

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The (timevvarying) ISAN captures the linear compression perspective since the hidden state can be seen as compression of the history { that sufficient generate the future {2z 17 j-t This definition also exhibits a general form of relational linearity (see c.g. Marconato et al. (2024); Hernandez et al. (2023); Paccanaro
Hinton (2002)) - any sequence tokens, not just those expressing relations in typical semantic sense, can be viewed linear operation applied to the embedding (the hidden state) of the history before it.
As mentioned, similar time- invarinttia model has been studied by Foerster al. (2017) for its interpretability, but our perspective differs because rather than viewing as separate architecture, treat it generative model that approximate modern language models. In fact, the work by Foerster et al. (2017) supports the usefulness this model as a theoretically tractable surrogate, although far from state of the art, scales. We present a new motivation for studying this model by proving that the extended logit matrix being low rank is equivalent to a language model being expressible as a time-varying ISAN.
achieve relatively coherent language modeling at small
Theorem 4.3 (Equivalence between Low Logit Rank and Time-Varying ISAN). Let M be lan- guage model over sequences of length T. Then M is expressible as a time-varying ISAN with hidden dimension d if and only if the logit matrix L (2*, -ST-)) has rank at most d for all S T.
4.2 REPRESENTATION POWER
We show that ISANS represent variety of simple architectures and languages. Specifically, we show that they can represent linear SSM layers, and can express copying and noisy parity. The formal statements and proofs deferred Section D.2.
4.3 PROVABLE LEARNING GUARANTEES
Given that ISANS can express noisy parity, and the standard cryptographic assumption on the hard- ness of learning noisy parity, it is computationally hard to learn an ISAN from samples (see The- orem D.13). To circumvent this, we consider a logit query model, where the learner can query the logits for the next token given any history. This model mirrors settings for practical model steal- ing attacks e.g. (Carlini et al., 2024), for stealing the embedding dimension and output embedding matrix for production language models and also connects to classical and modern works compu- tational learning theory (Angluin, 1987; Mahajan et al., 2023; Liu & Moitra, 2025).
Our main result is the following theorem, which shows that given logit query access to time-varying ISAN, we can efficiently learn a vime-varying ISAN that approximates it well in TV distance.
Theorem 4.4. Given logit query access to an unknown time-varying ISAN M over Σ with hidden dimension at most d, there is an algorithm that uses poly (d, [E], T. 1/c)) runtime and queries and returns description of a time-varying ISAN M' such that E[Drv (M, S
CONCLUSIONS
Our paper develops a simple but mathematically justified framework built the low-rank structure of the extended logit matrix and demonstrates its empirical predictive power across different models, data, and tokenizers. We believe this low-rank generative viewpoint will provide general model- agnostic foundation for understanding, probing, and steering language models. While some of the empirical demonstrations points to many promising future directions, both for new understandings and practical applications, which we discuss in more detail in Section
this paper such as LINGEN are proof-of-concept, we believe this lens
ACKNOWLEDGEMENTS
AL was supported by Miller Research Fellowship. We are grateful to Adam Block, Akshay Krish- namurthy, and Ankur Moitra for helpful comments.
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ETHICS STATEMENT
We view the results in this paper as fundamental research and do not see ethical concerns. While some of the insights may eventually lead to attacks jailbreaks against L.MMs, discovering such attacks is an important part of the research process, and we believe that the benefits outweigh the concerns and that the insights in this paper will help towards better defenses well.
REPRODUCIBILITY STATEMENT
Full experimental specifications are described in detail in Section B, and full proofs of our theoretical results are given Section We plan make the code for our experiments available.
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A RELATED WORK
Low Rank Structure in Language Models There has been significant work understanding the implications of the low rank structure in the next-token logit matrix. In particular, the output em- bedding matrix constrains the vector of next-token logits be in a space of dimension equal the hidden dimension (rather than the vocabulary size) and thus limits the family of conditional distribu- tions that be realized. This phenomenon, often called the softmax bottleneck, was formalized in Yang et al. (2017). The implications of this for language modeling are also studied in Press & Wolf (2016); Kanai al. (2018): Finlayson al. (2023). Furthermore, recent works study this softmax bottleneck from different perspective, showing that it leaks information about models even behind black-box APIs and results vulnerabilities to model stealing attacks (Finlayson al., 2024; Car- lini et al., 2024). However, these works all focus just the next-token logit matrix, whereas our framework allows us to reason about generating much longer sequences.
There have also been many works that observe low-rank structure in the weight matrices of machine learning models, dating back to the early days of deep learning (Denil et al., 2013), and this has been used to understand (Aghajanyan large models e.g. LoRA and its variants (Zhang et al., 2023; Hu et al., 2022). While similar in philosophy, approach is very different because it is model model-agnostic, does not involve looking the weights, but instead finds low-rank structure directly in the logits over sequences of tokens.
al., 2020) and develop efficient methods for fine-tuning
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Linear structure in word embeddings has been observed in well-known works (Mikolov et al., 2013; Goldberg Levy, 2014; Levy Goldberg, 2014) and also to some extent for sentence embeddings (Zhu & De Melo, 2020; Bowman et al., 2016; Li et al., 2020). The linear representation hypothesis is, informally, the idea that concepts are represented linearly as directions in some representation space. Park et al. (2023) propose formalization of this notion for language models, showing that certain concepts for the next token, such as gender, can be represented linearly in either the logit vector the last hidden state. There have also been many works, that broadly fall under mechanistic- interpretability, on probing and steering using the linear representations in internal layers of the model e.g. Elhage al. (2021); Meng et al. (2022); Hernandez al. (2023); Nanda et al. (2023); Turner et al. (2023); Todd et al. (2023); Hendel et al. (2023); Li et al. (2023); Geva et al. (2022). Compared to these aforementioned works, our framework only uses the output logits, but also allows us to reason about generating longer sequences of tokens (beyond just the next token). It is an exciting direction to try to extract features and perform interventions through the representations provided by our framework.
Related Theoretical Models Linear dynamical systems form the basis of control theory and are also the key building block in modern state-space models. Given the vast literature in control theory, we refer the reader to (Kailath, 1980; Chen, 1999) for classical treatment and (Hazan & Singh, 2025) for a modern treatment. A sequence of works over the past few years (Gu et al., 2021a; Gu & Dao, 2023; Katharopoulos et al., 2020; Dao & Gu, 2024; Gupta et al., 2022; Gu et al., 2021b) has led to the development modern state-space models, introducing additional twists such as selective gating (which is related to the loken-dependent transitions use) and connections to linearized transformers. Further connections to linear dynamical systems have led to new theoretically mo- tivated architectures (Agarwal et al., 2024). We believe our work presents a new perspective that is both theoretically and empirically justified, while maintaining the conceptual simplicity of linear dynamical systems.
In terms of simpler, mathematically tractable generative models, the ISAN model is related to weighted finite automata (Droste al., 2009) and hidden Markov models (HMMs) (Rabiner & Juang, 2003), except with the addition of a softmax nonlinearity to sample the next token. This non- linearity significantly improves its empirical performance on language modeling compared these earlier models (Foerster et al., 2017). There have been many earlier works studying the Icarnabil- ity of these classical models such as Mossel & Roch (2005); Hsu al. (2012); Balle et al. (2014). Query learning settings have been classically studied in the literature on learning automata and for- mal languages (Angluin, 1987), and more recently received renewed interest in the context of model stealing for large language models (Mahajan et al., 2023; Liu & Moitra, 2025). Compared to these works which study classical models (e.g. HMMs), we believe the theoretical work here takes another step closer to modern language models.
B EXPERIMENTAL DETAILS
B.1 ADDITIONAL EXPERIMENTAL DETAILS FROM SECTION 3.1
B.1.1 SETUP FOR EXPERIMENTS
Choice of datasets and models. We considered the following choices for the autoregressive lan- guage model M and the dataset D:
1. The model M was chosen amongst OLMo-lb, Olmo-7b (OLMo et al., 2024), Llama-1b lb (et al., 2024), Mamba-1.44 (Gu & Dao. 2024), Gemma-lb (et al., 2025). Moreover, as a baseline com- parison, we considered the checkpoint for OLMo- at time step of training.
2. The dataset D was chosen amongst: the wiki, arxiv, starcoder subsets of the olmo-mix-1122 (OLMo et al., 2024) training dataset: the math subset of the dolmino-mix-1i 1124 dataset (OLMo et al., 2024); and the c4 dataset (Raffel et al., 2020).
Next, detail the construction of the sets H,F from dataset D. The datasets D used each consisted of collection of many sequences, i.e., we have D = (1))) ,..., (N)) for some large N.
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Construction of histories. To generate a set H of histories of size n, we fixed parameters minn fmax- We first chose a random subset of elements of D of length least fuax (as measured by the number of characters). For each sequence y E S, we selected uniformly random contigu- ous subsequence of y of length max. We then chose e' ~ Unif[f fmax], and added sequence V1.6 H. (rounded to the nearest full word). We opted to measure length (and perform truncation) with respect to characters (as opposed to tokens) since we wish to use the same set of histories (and futures) for different models, which typically have different tokenizers.
Construction of futures. The set of futurres F was generated identically to the set H of histories, with the exception that in the context of the previous paragraph, Velly (where lyl denotes the length of y) was added to F.
Computation of the exponent To compute the power law exponents shown in Figures 2 and 7, we used least-squares regression, as follows. Given a matrix L E IR" (where n S m.) let its singular values, arranged in decreasing order (excluding the largest) be 01,..., -n-1- We performed 2-dimensional linear regression with covariates (log(i/n), log((n - i)/n]) and labels log og for i E [n- - 1]. Adding the covariate log((n - -n)/n) yielded a better fit as it accounts for the precipitous drop in as i approaches the number of rows (when the power law as described in Section 3.1 ceases to describe the decay of the singular values o,). The resulting coefficient for the Io - i)/m n) covariate was always very small, meaning that its effect was negligible for i < n.
Moreover, in the regression, the data point corresponding to index was weighted by 1/i. We chose this weighting (as opposed to, say. uniformly weighting all data points) to account for the logarithmic scale of the covariates terms of their dependence on
B.1.2 DETAILS FOR SINGULAR VALUE PLOTS
Recall that for an integer k. LM (M, denotes the sub-matrix obtained from Las CH,,H, F) by select- ing, for each f E F, the columns (f.z) indexed by tokens & which are one of the & most likely next tokens following f. with respect to M. Figures 2 and 7 show the singular values for logit matrices LM,, F) for fixed sets H, of histories and futures of size approximately 10000, as generated according to the procedure from Section B. For this procedure we chose fiin 50, L 200. The figures also show the singular values for 50,50(H4 F.), where H. (resp., F,) contains the first [HI/Vi (resp., [FI/Vā, elements of H (resp., F), for i E {2,4, 4, 8, 16). Thus the number of ele- ments (M,, Fi) is roughly 1/i that of L F). Singular values were normalized by the square root of the number of elements of each matrix.
Next, Figure shows the singular values of (M, when M was OLMo-Ib and H.F were obtained via the procedure described in Section B.1.1 when the dataset D was taken to be arxiv, st arroder math, c4. We remark that for these plots the sets H, F were taken to be of size only 4000.
1.1.3 DETAILS FOR KL DIVERGENCE PLOTS
Figures la, 8 and 10 show the average KL errors between the logit matrices considered in Figures 2, 7 and respectively, to low-rank approximations. Low-rank approximations were computed using an approximate singular value decomposition (via torch. torch.svd.lowwe.... svd.lowrank) and then zeroing out all singular values apart from the largest r ones. (Note that, for an exact SVD, this procedure gives the closest rank-r approximation in Frobenius norm.) Since we are considering the submatrices L.,, F), these KI, divergences are taken with respect to distributions restricted to the 50 most likely tokens per future.
The dashed line at the top of the figures represents the following rank-1 baseline: for each future f. we simply use the next-token distribution for f. i.e., Prw| f']. to approximate all entries (i.e., rows) corresponding to f. We remark that this approximation is in general not the optimal rank-1 approximation to the logit matrix (with respect to Frobenius norm).
For some of the larger datasets, we used buffer size of at least 10 to stream the dataset, so technically
the set of histories was not uniformly random subset.
'In particular, we generated sets H1 F of size exactly 10000 and then filtered out O-length futures, of which there were approximately 60.
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Figure 7: Singular values for 500 (H, F) for various models M; see Section B. 1.2.
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Figure 8: Average KI. divergence to low-rank approximation for various models M; Sec- tion 1.1.
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B.1.4 SUPPORTING THEORETICAL RESULTS
Below collect couple of standard facts which we use in our discussion of the experiments. For completeness, we provide their proofs (which are standard).
Fact B.1 (Phase transition in singular value decay). Suppose I, E IR with S m, and its singular values 01,...,00 On satisfy o, x i-, ff some constants C, a Then:
If a > 1/2, then for any r
N, there is a matrix of rank v for which [T. - All 1 O(r2-a). (Here) || || represents the Frobenius norm.)
2. If 1/2. then any matrix A of rank r S 2nd waς satisfies | AlF 2(11
In the above statements, the 0(),9(() hide absolute constants.
Proof. The fact is standard, but provide the proof anyways. Let suppose that Co. C1 >> 0 are so that Co i-a & o, S Ci i-a for all i.
Case 1: a > 1/2. First suppose that > 1/2. We may write the SVD of L. Σ 0; uay, T where {u;} and } are orthonormal bases of R² and R", respectively. Note that |L||
V of 2 Co Ω (220-1)
Fix any N, and let A Eim o..iy Then |L Al|F v V of -0 It follows that
r1 2i 2a-1
| |1-
Ci Co <0(G) r rd--aa
|
where the O(+) hides dependence on an absolute constant, as desired.
Case 2: a < 1/2. It is standard fact that for any r, the best rank-r approximation to a matrix L given by truncating its SVD at rank-r, i.c., for any rank-r matrix A, we have that
n 1-120 11-2a
| AllA 2
o z
Co
2a -
But |||||= S S 0. 0 desired.
) 1-2a
n1 nl-
meaning that for
S
we have
LL 2 ((Co/C1), [[
2470 2n7
Fact B.2. Consider sets H. F of sizes n, m respectively. Then any matrices A, E (xx(Fx)) holds that
Σ FEF
DKL Ds (A),))))(()))))))))))))))))))))) < ||A-A111.
Proof. It follows immediately from the definition of average KL divergence that it suffices to prove following fact: if 0, E Rdd, then letting p = softmax(v), p' softmax(v') (so that p, e Ad), then we have DKL (pllp') < ||l - 1'113. To prove this fact, let define F(v) log Σd e": Then direct computation yields
DRL DKL(plpp = F(v) - F('')(())))) (VF()), v).
Next, for any I E Rd. for q = softmax(x) we have V2F(z) = diag(g) "9 - La i.c., F is 1-gradient Lipschitz. But this yields that the right-hand side of Equation (1) bounded above by ||e v/133. as desired.
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Figure 11: ((),, (b)): Low-rank approximability of (M,, (c), (d): Low-rank approxima- bility of CM.50 F); see Section B.1.5.
B.1.5 ABLATIONS FOR THE PARAMETER
Recall that our experiments exhibiting approximate low-rank structure of the logit matrix actually apply to the sub-matrices M FH, described in Section 3.1, for k 50. In Figure 11, we present two modifications to this approach (all with DLMo-11)) in Figures la and 1lb we measure the low-rank approximability of 2,, 200(7 H., [H1, |F] to be of size approximately 2000 in Figure 1a and to be of size approximately 5000 for Figure 11b.
F), for sets H, F. Due to memory limitations, we took
Next, in Figures 1lc and l1d, we measure the low-rank approximability a submatrix 5..50 (H, obtained from Ca (H. F) by sampling randomly 50 tokens per future and selecting the correspond- ing columns LM (H, F). For these figures we used H. F of sizes approximately 4000.
All of the aforementioned ablations show similar evidence of low-rank structure similarly to the logit matrices (?, F). We remark that, in general, as the size of H, F are increased, the fitted power law exponent describing the decay of the singular values tends decrease slightly, perhaps because the effect of the slight "hump" for the first few singular values is diminished. This may explain why the exponents in Figures a and 1 1c slightly smaller than in Figure 7a.
B.2 SUBSPACE COMPARISON
Figure 4 displays the principal angles between low-rank approximations of SM, and L..30 Fnoosense) where the model M was OLMo-1b, H. were generated from wiki as de- scribed in Section 1.1.1 with [7| = 10000 and where F, Fnonsense were each of size approximately 5000.7 Low-rank approximations were computed by truncating (approximate) singular value de- compositions, as described in Section 1.1.3. The dashed lines show the principal angles between pair of independent uniformly random subspaces of the indicated dimension. is evident that the
We started with sets F, Foonsence size 5000 and then removed a small number of O-length futures.
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of Angl
244
720
Au
Dhant 2441
fandom Dani: 2441
Tardon Ray: 2441
(a) Gemma- 1B vs Mamba-1.44
(b) Gemma- IB vs Olmo-78 7B
(c) LlaMA- IB vs Gemma- 1B
Tanic
Pannc 340
Coree woo.eraaai
Nundin Banic
160)
Jum .021
Dumi: LCC)
(221
Garbon Part: (221
(e) LlaMA-IB vs Olmo-1B (ckpt
(d) LIBMA-1B Mamba-1 1.4B
(f) LlaMA-1B vs Olmo-7B
Namic
Kunu
Oph Oohestenaa
Naric 244
(lrck 1991
Jaien 1991 Nardon Bani 2221
sance 2441
fandom Jart:
Edo
(g) Olmo-1B vs Gemma- IB
(h) Olmo- IB LlaMA- IB
ii) Olmo- vs Mamba-1.44
Nanc 61
Nunc 120
Nunden Sumi: 30C
pun (9)
Nurdor Suvi: 244)
2441
1221
Aam. 1221
(I) Olmo- (ckpt 0) vs Gemma- 1B
(j) Olmo- 1B vs Olmo- BB (ckpt 0)
(k) Olmo- IB vs Olmo-7B
Fami:3 33
Demo 33
Teric Nanc
of Prepet
Nerdom 322
Kas: 2M
1891 fancer (ank 2001
lardon Janic LoC
landen bani 2021
Kirkrn Pave (98)
conve
landon
(m) Olmo-IB (ckpt 0) Mamba-1.4B
(n) Olmo- (ckpt 0) Olmo- 7B
(o) Olmo-7B vs Mamba-1
Figure 12: Principal angles between column spaces of logit matrices across all pairs of models; see Section B.2.
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History
Closest history
Investigations personnel in 1927. The name, derived from eighth letter of the Greek alphabet, appears to have been first used on a 1946 Argentine government chart following surveys
(BIT) was created on 9 December 1898, in response to the Bombay plague epidemic 1896. It was created
on Honest Jon's capitalised on this with Theo Parrish, Rashad & Spinn, Ricardo
for Nelly Furtado, N-Dubz, Nina Sky, Lil' Flip. Noel Gourdin, Starboy Nathan, Shawnna, Jimmy Cozier, Alison Hinds, Jazmine Sullivan, 20, Cory Lee, Chris Webby, City
Shiroro Airfield is an airstrip serving the village of Shiroro and the Shiroro Hydroelectric Power Station in the Niger State of Nigeria. The runway is
Canadian Forces Base located immediately south of the town of Chatham, New Brunswick, Canada. Parts are now operating Miramichi Municipal Airport since 1974 with partial runway
to numerous publications, and supported the development of range of new PES schemes including the BioCarbon Fund at the World Bank and the Mexican PES
as mitigation bank providing ecosystem services to the public in the form of Environmental
Thereupon, Gangaram slays out Janardhan Seth, perceiving the dirty deed angered Rambabu reaches Janardhan Seth's s
stealing the idol's jewellery and his wife is accused of being witch. They are killed by mysterious person after which Chandni's 's father Yash Narayana Vashisht becomes the Mahant. Dev escapes
process of removing entry and price restrictions on airlines affecting, in particular, the carriers
stated model. maintains close ties to other Green parties
On Spitfires. Scotland was in range of Nazi Germany's 's long-range bombers and reconnaissance aircraft. The Luftwaffe's
minor improvements that increased their anti- unti-aircraft capabilities. Their crew numbered 228 officers and enlisted men. The floatplane of the 1910s produced by Flugzeugbau Friedrichshafen.
U-boat built for the Nazi Germany's "Kriegsmarine" for service during World War II.
European works, classical Chinese works and Nôm works into Quoe Ngu modern appointed Assistant Chaplain at Geelong Grammar School from 1959 to 1961, vicar of Romsey and Sunbury with Lancefield from 1961 to 1964
supported the creation of the Empire of China and the 1917 Manchu
the Young Women's 's Christian Association of New Zealand. She was a member of its Dunedin board from 1930 until 1944, Dunedin
Table 1: For values of H (left column) we display the history // E H. * minimizing the distance between the respective rows of the logits matrix, i.e., ((h), La (Mr(())) F) Examples were not cherry-picked (they correspond to the first 10 examples in the set H described in Section 3. as obtained from wiki).
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column spaces of the logit matrices have significantly more overlap than what would be expected from random subspaces.
In a similar manner, Figure 12 shows the principal angles between low-rank approximations of (H, F) and L (ar,55(( (H, for each pair (M, M') of models described in Section B.1.1. For these matrices, we used the same sets H. F deriving from the wiki dataset were used in Sec- tions 1.1.2 and B. 1.3. Note that all pairs of models (M. .M') show significant overlap in the column spaces of their logit matrices, with the exception of when either M or M' is the untrained variant of OLMo-Ib. lb. This is unsurprising, given that the latter cannot be expected to capture any semantic information in sequences. Interestingly, there is still bit more overlap than would be expected from random subspaces. We believe this may be because of the nature of the transformer architecture: for instance, two sequences are very close (e.g., differ in single token), then it reasonable to expect that the next-token distributions induced by even a transformer with random weights are somewhat close, due the nature attention. We leave further investigation of this point future work.
B.3 LINEAR GENERATION
The experiments using LINGEN in Section 3.3 make use of OLMo- lb and generate for total of 15 tokens. Below we provide details pertaining how the coefficient vector v was chosen for each of Option 1 and Option 2 described in Section 3.3:
Option 1. For this case, we chose 40000 histories and 10000 futures from the wiki dataset, as described in Section B.1.1. We chose the lengths Emin Lmax 30, but now measured the lengths with respect to tokens (not characters), since only intend use the resulting histories and futures for a single model. As the previous sections, because the token space E is extremely large, cannot perform the regression deseribed in Section 3.3 on the entire logit matrix LM F), so we instead used the submatrix CM,50(H, FF as described in Section 3.1.
Option For this case, we generated sets H. of historics and futures as in Option but then randomly permuted all tokens amongst all elements of each of H. and F, which yielded sets nonsensse Fnonsense
The remaining details the experiment are identical to Option 1.
Baselines. In Figures 1b and 6 we compare the performance of LINGEN to the following baselines:
Single-token version of LINGEN: this may be viewed using LINGEN but where the matrix M0 (M, F) was replaced with (H, (Null)), i.e., the full logit matrix where there is a single future, namely the empty future.
Generation from a short context:
we generate from OLMo-1b using a context window 5 tokens (i.c., we mask out all tokens more than units the past).
Intermediate training checkpoint: we generate from OLMo- lb using the checkpoint Sttggl/sttp110077399-tkmo4001 which corresponds to the end of Stage- pretraining.
Algorithm LINGEN Algorithm
1: Input: Language model M, histories H Σt, vector u R
2: Input: Parameter m (number of new tokens to generate)
3: function LinGen (H, v, m)
4: for to = 1,2, do
5: Compute logits, = TT. LM,H,, (215-1)). D (HH (21--13) the logits matrix
where the set of futures is the singleton sequence 21:1-1.)
6: Sample next token 2, ~ softmax
7: end for
8: Return: the sequence (e.....,2m!
9: end function
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LINGEN
LINGEN with single-token logit matrix
Third Sea Lord was given the command upon taking leave from the Admiralty, He hoisted his flag in "Bacchon" (1874), which was in the Mediterranean under Rear Admiral
Third Sea Lord was given the command upon taking leave from the Admiralty. He hoisted his flag in "Bacchusa in 1881 I and 1892 compilation also showed that the
in 2007 in Rajasthan, India Krishna Bhatt has been twice the recipient of the Award for literary Excellence and has been Ghana, Switzerland,
in 2007 in Rajasthan, India Krishna Bhatt has been twice the winner of the Golden Mapuhal Trophy at the General Bob badi
and Katy Gibson in "Gigi" had the world's best-known actors and pop singers Peter O'To
and Katy Gibson in "Gigi" and "Diva" were produced, and they were thus of the
2018 including Jago Wali Raat and her first devotional song Tor Dita Lalan Nu. For the song God Save India, she uses traditional instruments like Bhajan
2018 including Jago Wali Raat and her first devotional song Tor Dita Lalan Nu. Rebecca and Mr. Costa confess to each other that they were falling in
KSTQ (93.5 FM) is a radio station licensed to Stuart, Oklahoma, United States. The station is owned by KFBL Radio and is currently simulcasting KXLP
KSTQ (93.5 FM) is radio station licensed Stuart, Oklahoma, United States. The station is locally called "The Rock," is currently operated by Axion and E
Sample generations from LINGEN (Option 1) and single-token baseline. Prompts shown in gray and italicized, and continuations are shown in black. Samples were not cherry-picked, i.e. the first generation from each prompt was selected. The generations from LINGEN (left) all read well and make clear use of context in the prompt; in contrast, while generations from LINGEN with the single-token logit matrix (right) typically are reasonable for the first token few, they become derailed soon thereafter.
Table 2:
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LINGEN generations with 7 nonsense
Third Sea Lord was given the command upon taking leave from the Admirally. He hoisted his flag in "Bacchill" confusing manoeuvre that involved making the windward course.
Third Sea Lord was given the command upon taking leave from the Admiralty, He hoisted his flag in "Bacch" for the first time. After Sapp didn't take part in Third Sea Lord was given the command upon taking leave from the Admiralty, He hoisted his flag in "Bacchel" and joined with her and the USS Vixen, a
in 2007 in Rajasthan, India. Krishna Bhatt has been twice the National Head of Dairy Labelling 2004 and 2014. He
in 2007 Rajasthan, India. Krishna Bhatt has been twice the national shortlisting for the International Youth Olympic Festival (TYOOF) in 2007 in Rajasthan, India. Krishma Bhatt has been twice the author of a book entitled "Cheerbato Jaataa Aap
and Katy Gibson in "Gigt". Consists of: -Irb- lead -rrb- and Katy Gibson in "Gigi", for which she received a nomination at the 2020 Jogja Music and Katy Gibson in "Gigi". Music career. addition to her television work, Bryant has released several
2018 including Jugo Wuli Raat and her first devotional song Tor Dita Lalan Nu Ghareba Hae. A new format was created by exposing a new
2018 including Jago Wali Raat and her first devotional song Tor Dita Lalan Nu, Ranthi Narayan Thakkar are known for their voice and
2018 including Jago Wali Raat and her first devotional song Tor Dita Lalan Nu Ram" She made her film debut opposite Satish Kekar in Raj
KSTQ (93.5 FM) is a radio station licensed to Stuart, Oklahoma, United States. The station is 24/7 transmitter reports that broadcasts to Tulsa, Page,
KSTQ (93.5 FM) is radio station licensed to Stuart, Oklahoma, United States. The station is locally owned by Quad Cities Radio and maintained by Educational Insights. In KSTQ (93.5 FM) is a radio station licensed to Stuart, Oklahoma, United States. The station reading of Victim News radio station licensed to Tulsa, Oklahoma broadcasts
Table 3: Sample generations from LINGEN and single-token baseline with Hnonsense (i.e., Option
2), for the same target prompts as in Table 2. Prompts are shown in gray and italicized, while LINGEN's s generations are in black. While not all of the generations are natural continuations of the target prompt, most of the generations show clear use of information from the prompt.
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C COMPARISON TO NATURAL BASELINES
In this section, we briefly discuss comparison of the low rankness of the extended logit matrix to natural baselines. First, let us consider a random model for comparison. Recall that the extended logit matrix F)) is a [H]] x [F] |>| matrix. A matrix of the same dimension with i.i.d. Gaussian entries, denoted by G (i.e., Gūj (0, 1)), has a well-understood asymptotic spectrum given by the Marchenko-Pastur law. Further, non-asymptotic bounds on the singular values of G are also known (See e.g. (Rudelson & Vershynin, 2010)). In particular, the results suggest that the singular values of G are all of the order ((vn) where n max [F1 |EI} with even the deviation of the smallest singular value from v being of the order ((n This indicates that the singular values of G do not decay rapidly and are all of the same order and hence is not approximated well by a low-rank matrix.
More, interestingly in order to observe power law decays in the singular values, as noted empirically in Section 3.1, one needs to consider matrices with dependent entries. An interesting future direction suggested by our work is to potentially find interesting random matrix models that can be both used explain the power law decay in the singular values of the extended logit matrix, while also helping us understand the underlying structure in natural language.
second natural baseline compare the low-rankness of the extended logit matrix that arise purely from the low-rank structure of the single step logit matrix. To flesh this out consider, the matrix
L = Ly (H, for H = E¹ and F = Null and set i = L (H, F) for H = E and F = 24/2 The choice of t/2 was made for simplicity, and the argument can be easily extended to other splits of t. Now, from the argument regarding the low-rankness of the single step logit matrix, we have that (I.) < d but for the sake of argument let us assume that 1. This indicates that the rows of are all multiples of each other, again for simplicity assume that it is constant multiple of the first standard basis vector. Understanding the struture of L tells us that the rows of L are concatenations of the rows of corresponding to splits of the history into two halves. But, since the coefficient corresponding to each row of I, was arbitary, the rank of Í. corresponds to the rank of an arbitrary matrix of dimension |>]+/2 x |2]6// which is exponentially large in Thus, the low-rankness of the single step logit matrix does not imply any non-trivial upper bound on the rank of the extended logit matrix. In particular, this can be seen as strong argument for the claim that low rankedneessss of the extended logit matrix is an extremely surprising phenomena that points towards the presence of interesting structure natural language and models thereof.
D DEFERRED PROOFS OF THEORETICAL RESULTS
As reference, first state the definition of an Input Switched Affine Network (ISAN) by Foerster et al. (2017). Note that the difference compared to time-varying ISAN in Theorem 4.2 is that the transitions depend only on the current token but not on the timestep.
Definition D.1 (Input Switched Affine Network (ISAN)). An Input Switched Affine Network (ISAN) is defined by matrices Ag E I for each z E Σ and a matrix B E RRExd and an initial state To E Rdd defines distribution over sequences of tokens at each timestep 1. = 1,2,....
Sampling: token 2, sampled from softmax( BBx+-1).
State Update: the hidden state is updated as It- = At-
First, we show an equivalence between an ISAN and time-varying ISAN up to a factor of T' in the hidden dimension.
Fact D.2 (Reducing& Time Varying ISAN to an ISAN), For any time-varying ISAN with hidden dimension d that generates distribution over sequences of tokens of length T, there exists an ISAN with hidden dimension Td that generates the same distribution.
Proof. Let the initial state of the time-varying ISAN be 2n E Id and its transition and observation matrices be Ag.t Raxd B, E Exd for E Σ, E (T). To express this as (time-invariant) ISAN, we build a T Tdddimensional hidden state consisting T separate d-dimensional blocks. We
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can then aggregate the transition and observation matrices across the different timesteps by defining for each token E Σ,
0
0
0
A
A
A22
0
A
B
[1 B2 ... BT)
We Iet the initial state be O......... i.c. To on the first coordinates and everywhere else. It immediate by construction that this ISAN is equivalent the time-varying ISAN because each timestep f, the hidden state It of the original time-varying ISAN is simply stored in the + 1st d-dimensional block and is moved to the next block after transition (the state becomes identically 0 after the last transition). Thus, this ISAN generates the exact same distribution over sequences of length T, desired.
D.1 EQUIVALENCE BETWEEN TIME- VARYING ISAN AND LoG LOGIT RANK
This subsection is devoted to proving Theorem 4.3. We first show one direction, that low logit rank implies expressibility as a time-varying ISAN.
Lemma D.3 (Low Logit Rank Implies Time-Varying ISAN), Let M be a language model such that for each E (T), the matrix Las (E', EST (recall Theorem 2.2) has rank Then there a time- varying ISAN of hidden dimension d that generates the exact same distribution as M on sequences of length T.
Proof. For each length B..,ghht-T....-111 let C. E' be (multi)set of sequences with |8(1| d such that the rows of LM (S((), -ST- span the row space of (2r -ST In other words, this saying that the rows indexed by ((t) span all of the rows indexed by sequences of length t. Note that such a set exists by the rank assumption - for t = there only one row, and we allow for arbitrarily choose to duplicate some of the elements that
multiset if |E9] & d then
|860| d exactly.
For each timestep E [T] and token 2 E E we can construct a matrix Az,t E Raxd such that LM( EST As, SST-1)
where on the LHS, S(t-1) o 2 denotes the set obtained by appending the token 2 to each sequence S(t-1), The above is possible because each of these are now length sequences in Σ and we assumed that the rows of LM EST-+) span all of these. Also, let B, E R-xdd be the matrix B, Las (S(t), (Null))T where here the set of futures consists of only the empty string.
Now we claim that the ISAN defined by Ar,;, B, for E Σ, t E [T] and initial state Tu (where é is the first standard basis vector) exactly samples from the distribution M over sequences of length T. We first show by induction that if is the sequence sampled so far after / timesteps, then the hidden state satisfies
x.7 L (S(1), EST-!) Lm ((21:). EST-1)
for 0, I, T. Note that there is only one empty string so the base case for 0 obvious. Now given the inductive hypothesis for t, we immediately have that for any choice of token 24+1,
x. (S(t)) 2111, EST-t-1) ((r EST-(-1)
just from Theorem 2.2, since the above is just selecting subset of the columns of the equality in (3). Now since x++1 = AT A combining with (2) gives
2 Ar Ca - Lu ((= (, EST-(-1)
completing the induction. Next, it remains to show that given (3) for all t, that each token is sampled from the same conditional distribution as M given the prefix a1.0. To see why this is the case, (3) implies that
2. B = I L (S(1), (Null) ) = Cas ((21c)) (Null))
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since this is just obtained by selecting subset of the columns in (3). Note that LM ((z1r)) (Null)) exactly the mean-centered next token logits conditioned on 21:6s and for both the underlying distribution M and the ISAN we defined, conditioned on the prefix 21st, the next token is sampled according to softmax(B;xx)) Thus, the ISAN generates the same distribution as M, completing the proof.
Now we prove the reverse direction that time-varying ISAN expresses distribution with low logit rank.
Fact D.4 Varying ISAN has Low Logit Rank), Let M be a time-varying ISAN with hidden dimension d that generates some distribution over sequences of tokens of length T. Then for any t E [T) the matrix (M, -ST (recall Theorem 2.2) has rank at most
Proof. For any sequence h Er, let 2ú the hidden state of the ISAN at timestep t, after out- putting the sequence of tokens h. Now consider any future f EST-(, Let the tokens of f be 241...... , Then the vector of mean-centered logits for the next token conditioned on a f. given by {Lar(2hh o f)}ees (recall Theorem 2.1), can be obtained by taking the vector
B ... A.411.4117
and then subtracting the mean from all of the entries. In other words, these logits are a linear function 24 and there is a matrix Ay Rxd such that Aya gives exactly the mean-centered logits (ZM(2/h f)rze.. This implies, by Theorem 2.2, that we can write
(x, (M,(E'ESTT EST-)=
Tha
[4],
AJ
...]
where the rows of the first matrix are indexed by all possible histories h E E' and the second matrix is a block matrix with blocks indexed by futures f E EST-1. The The above factorization implies that LM EST-t) has rank at most d, as desired.
Remark (Importance of Mean-centered Logits). Note that the mean-centering of the logits important for Fact D.4 be true. Otherwise, if we used, say, normalized logits, then these logits would no longer purely linear function of the hidden state In due to subtracting the normalizing constant which is not linear function of the entries (whereas the mean is).
Proof of Theorem 4.3. Theorem 4.3 now follows immediately from combining Lemma D.3 and Fact D.4.
D.2 TIME-VARYING ISAN REPRESENTATION POWER
In this section, we analyze the representation power of the varying ISAN, showing that it can represent variety of simple architectures and languages. First, we show that it can represent linear state space layers, the core building block in modern state space models (Gu et al., 2021a; Gu & Dao, 2023), We begin by formally defining the class of state space models that consider.
Definition D.5 (Selective State Space layer). A selective state space layer maps sequence of inputs U...... urT E RP a sequence of outputs y1...., . Yr E R2 and has a hidden state with dimension d. The initial state is E TRd and the state space layer operates according to the following recurrence:
A(u))a-1 + Bluz)u 3/2 C(u)) D(u)u,
where the matrices A(we) E Rdxdd B(u) E Rddx, C(()) E Raxd D(u)) E TRXP may depend arbitrarily the input W4 al timestep
Remark 2 (On Variants and Stacking of SSM Layers). This is general form that encompasses most definitions of a selective state space layer - implementations that are used in practice often enforce structure on the matrices, especially A. Earlier, simpler variants of the SSM used invariint layers where A, B, C, D are all fixed, independent T.
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State space layers are often stacked together- tiather--lttiee although with the above form, stack of state space layers can be represented as a single layer with dimension fd as long as the transitions matrices A(ue), ((u),, C(u)), (we) depend only on the external input (but not on the recursively obtained intermediate inputs to layers in the middle, see Hespanha (2018)).
Now not difficult see that an ISAN can represent a language model consisting of a selec- tive state space layer with softmax readout and no additional nonlinearities (where the tokens are embedded and then fed as the inputs we) as defined below.
Definition D.6 (Linear-in- in-sate state SSM). A linear- in-------te state SSM defines a distribution over sequences in Σ as follows. There is selective state space layer with input, output and hidden state dimensions p, 9, d. The oupuut tokens are obtained by sampling 2t ~ softmax(U ye) where U E Rx the readout matrix and the next input is obtained as u4-11 Ve(zz) where e(zx) E TR. denotes the one-hot encoding of 2, and V TRPx is the embedding matrix. The initial imput u1 is some fixed vector.
We now prove the following fact: Fact D.7 (Representing Linear-in-state SSMs), A Linear-in-state SSM with input, output, and hid- den dimension p, q, d can be expressed an ISAN with hidden dimension d + 2q +
Proof of Fact D.7. We construct an ISAN that at each timestep stores h4-1 (Iz, Clu)) (T-1) ((u,)),, 1) as its hidden state. Note that the dimension of this is d + 2g + 1. Now by definition, the next token 2, is sampled from (U vector inside the softmax is clearly a linear function of the hidden state. Next, once we fix the next imput u++1 is fixed and we have Z(41 = A( ((u++1 )a, + ((u+1) J441, and thus the next hidden state h2 (It+1, ((u+11 )2,, ((4+1)u44+ 1) is linear function of the previous
D(u)) and the
once u is fixed. This implies that the entire linear-in-state SSM can be expressed as an ISAN, completing the proof.
Next, we will also show that the ISAN model can express basic languages and algorithmic behaviors. We show that it can solve the task of copying, i.c. generating sequences of the form a where is a random bitstring. This task is a standard benchmark for testing the ability of sequence models to capture long-range dependencies and has been used to separate the power of different architectures (Jelassi et al., 2024).
First, we formalize the task of copying as being able to sample from the following distribution.
Definition D.8 (Copying). Define the n-bit "copyiig" distribution as the distribution over length 2n sequences of the form a where a E 10, uniformly random.
Now prove that ISANs can express copying.
Fact D.9 (Representing Copying). There is time-varying ISAN with hidden dimension n + 1 that generates distribution arbitrarily close in TV distance to the n-bit copying distribution.
Proof. Initialize the hidden state To to n zeros and the last bit is 1. Now for the first n steps, the output matrices ....... Bn are all (so the output uniformly random) and the transition matrices simply store the first n bits generated in the first bits of the state while maintaining that the last bit is 1. Now for the next n steps, the transition matrices are identity and the output matrices B Bzn are
0 Bu++ = OO
0 c ... 0 C/2 0
where the last column is always the same and the column with C' is the jth column and C can be chosen arbitrarily large. Then the jth bit of the state is O, then the output will be with exp(C/2)/(1 + exp(C/2)) probability and if the jth bit of the state is 1, then the output will be with exp(CC2)/ exp(C/2)) probability. Since C can be chosen arbitrarily large, this ISAN can generate distribution that is arbitrarily close in TV distance to the copying distribution.
O
*This result can also be applicd to a stack such layers, as long as we only allow the transitions to depend on the external input, but not the intermediate values.
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Finally, we show that the ISAN model can generate from a noisy parity distribution. Noisy parity fundamental problem that has been extensively studied in computational complexity, learning theory, and cryptography. In fact, the hardness of learning noisy parity from samples is a standard cryptographic assumption (Blum et al., 2003; Pietrzak, 2012). First, we formally define a noisy parity distribution:
Definition D.10 (Noisy Parity). A noisy parity distribution over (0, 1]n+1 is defined by a sequence y {0, 1) and probability p (0, 1/2) and is obtained by drawing E {0, 13" uniformly at random and outpulting (z, b) with b E {0, } defined as
(y, z) mod 2 with probability
b
1 (v, z) mod 2 with probability P
Now prove that ISANs express any noisy parity.
Fact D.11 (Representing Noisy Parity). For any noisy parity distribution over {0, (0,1)at1, there is vime-varying ISAN with hidden dimension 2 that exactly generates it.
Proof. We claim that for M being noisy parity distribution, matrices LM (10, 13", {0, 1]5*+1-() have rank at most for all S L S n n-+1 1. To see this, we simply observe that each of these matrices has at most two distinct rows - the only information in the first / bits that matters is the parity of the inner product with y restricted to these first bits. Now we apply Theorem 4.3 and this completes the proof.
The fact that ISANs can represent noisy parities, combined with the following standard crypto- graphic assumption, implies that it computationally hard to learn an ISAN from samples, which we state in Theorem D.13.
Assumption D.12. [Hardness of Learning Noisy Parity] There exists no polynomial time algo- rithm that given access to samples from a noisy parity distribution with hidden vector y can recover a vector y that is non-trivially correlated with y (say ly @ - < n/4).
Theorem D.13. Under Assumption D.12, there is polynomial time algorithm that given access samples from a time-varying ISAN can learn a distribution that is close in total variation distance.
D.3 LEARNING AN ISAN FROM LOGIT QUERIES
In this subsection, we present the details of our algorithm for learning time-varying ISAN from logit queries. First, we formalize the query model.
Definition D.14 (Logit Query). The learner can specify any prefix h e Σ* and obtain the mean- centered logits Las h|] for E Σ.
Observe that logit queries can simulate samples from the language model. We do this by repeatedly getting the next token logits and then taking softmax to sample the next token and then repeat.
Fact D.15. Given logit query access to M, given any history E y", we can sample a future f of any length from the distribution Pr
Definition D.16 (Slicing Notation). For any L L T, we will use the notation M[; 1) to denote the distribution of length & prefixes for sequences drawn from M.
Now we present and analyze the algorithm for learning from logit queries. At a high level, the al- gorithm tries construct "spanning" sets of histories and futures He, F, for t 0, I,I.,., 1. Ideally, we would be able to ensure that the matrix Las (H,, has rank equal to Car (x', ( which would mean that He, Fr in some sense cover of the dimensions. Unfortunately. this guaran- tee is not possible because the rank of L (E', T* could arise from some very rare sequences that will be impossible for us to find algorithmically. We can instead guarantee a weaker notion where we cover all dimensions that are not too rare. The algorithm, described by the subroutine in Algorithm 3, iteratively constructs the sets Fr by adding new sequences that increase the rank of LM (He, Fi) until no such sequences can be found. The key to the analysis is defining the right no- tion of coverage so as to ensure that the algorithm terminates in polynomial time (see Lemma D. 17). We then complete the proof by showing that with this guarantee, we can construct an ISAN that is close to the true distribution M in TV distance.
yST-t)
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Algorithm 2 Learning from Logit Queries for (time-varying) ISAN
1: Input: logit query access to time-varying ISAN M
2: Initialize H, {y'} } for an arbitrary y Σ for
....,,.. T- (where y" Null)
3: Initialize Fr {Null} for = 0, o.,,...,T -1
4: Set N 4c log/ dff/c)
5: Set (Hr),, {F+} <- CompleteSpan as ((Hi)r. (Fi).,N)
6: Set initial state to be To E R2 as 2o =
7: for e 1],2 E do
8: Solve for Az,t E ( such that
12 Agt Fi)
9: end for
10: for E [T' do
11: Set B₁
LM (H--1, (Nul),
12: end for
13: Pad matrices Az, B, to appropriate sizes (d x d, [X] x d respectively) by adding rows and columns of zeros
14: Output: ISAN with parameters To, Ag,, B
Algorithm 3 Complete Span
1: Input: logit query access to time-varying ISAN M
2: Input: Sets F, for
1,....,77
3: Input: Parameter N
4: function CompleteSpan (1H4):, N)
5: Initialize Incomplete True
6: while Incomplete do
7: Set Incomplete False for / 1] do Construct S, by taking N i.i.d. samples from M[:
8:
9:
10:
Ra (H,, < Rank(Car U H UUS,FF FUUD F++1)) then
11:
Set Incomplete True
12:
Find elements h E H4-1 EUS, and
such that Rank (ank(LM (H4 U F, = Rank(L F.)) +
13:
Set H4 H4 U (h) and F, (.)
14:
end if end for
16:
end while
Return: {H4)., {Fi)
18: end function
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Lemma D.17. Algorithm 3 terminates within dT iterations of the while loop and with - dT (1- (2(2)) probability, we have the following property: for any [T 11, if we draw h M then with e2T()))) probability over the randomness of h,
Rank(L M (H,,FF)
Rankk(LM 1 EU F.US.
(4)
where Ht-1 o denotes the set of all possible sequences obtained by concatenating some element of H4-1 with some token in E.
Proof. Note that by induction, we always have [H4 - ( E {0, and [F
Mank(CM(H,,, Fi)) {0, (the off-by-one is just because the initial matrix could have rank 0). Note that since M is generated by time-varying ISAN, by Fact D.4, the rank is always at most d. Also in each iteration of the while loop before termination, the sizes of He Fr increase by for at least one t. Thus, the algorithm can execute at most dl iterations of the while loop before termination.
Next, to prove the second part of the lemma, whenever the desired property doesn't hold for some t E [T 11, then with probability at least (1 </(2T)) the check in the while loop will fail and the flag Incomplete will be set to True. Thus, union bounding over every time we check the rank condition, the overall failure probability is at most d (1- (0e(((11T)) e
Now can complete the proof of Theorem 4.4.
Proof of Theorem 4.4. First, note that the ISAN we construct in Algorithm is well defined because whenever Algorithm terminates, we must have
Rank(C (H4, Fi) Rank(C U
and thus the matrix Ag,t must exist because the rows of (HH, must span the rows of LM AMrHHe (H-1 o Ft). Also note that all of the operations in Algorithm 2 can be implemented in poly(d, [I, time and queries.
Now assuming that the property Lemma D.17 holds for the output of the CompleteSpan step, we claim that we have Drv M') s 0.5c. For each f, let the good set G. C E' be the subset of histories h for which (4) holds.
Now sample token by token according to the ISAN we learn with parameters To, Ag,t, Be. At timestep & 0, by construction, we have
To
= Fo)
and our current string is Null. Now assume that timestep I, have sampled the / tokens 2ı. and the current hidden state & satisfies
f LM (H,, = Lss F1)
Then since Null Fs, we get that softmax (B,G.) exactly samples the next token with the correct probabilities. Let the next token be 24 If 2112 G., then the above equality also implies
E T LM 2441 o F(-1) La 241 F++1).
which is the same saying
Fe41) Tú LM F(41)
= A Las (Het),
= - T (Heri,
In other words, by induction, as long as all subsequences remain in the good sets 91.... Gr,, then ev- ery token is sampled according to the correct conditional distribution, matching M. By Lemma D.17 and the definition 0.5c. The way 1 - 0.5c and thus the overall expected TV distance between M' and M is at most e, as desired. Li
the good sets, the probability we ever sample outside of a good set is at most N ensures the conclusion of Lemma D.17 holds with probability at least
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D.4 GENERALIZATION BOUNDS
In this section, we show a generalization bound for when the LinGen algorithm can guarantee high accuracy. We assume that have fixed the language model M, have target history ho, and we will approximate ho as a linear combination of "basis" histories H. = (hi,..., h. We then generate & tokens according LinGen (Algorithm 1).
Lemma D.18 18 allows us to relate the linear regression error when we draw futures from an arbitrary distribution the KI divergence between the LinGen distribution and the ground truth distribution for continuations of /o, as long as we have the matrix ordering inequality in (5). The main point that (5) and (6) (the regression error) can be tested empirically from samples because of matrix concentration (see Fact D.20). Lemma D.18 then shows that if these quantities are sufficiently small, then we can bound the KL divergence between the true distribution and the one obtained by generating according LinGen.
While our theoretical bounds are not tight enough to directly apply, we still believe this a worth- while conceptual point that generalization can be bounded in terms of some testable quantitative "subspace overlap" that related to the subspace overlap for different distributions of futures F, F discussed in Section 3.2. We also discuss below how expect tighter bounds to hold in practice due to better concentration than the "worst-case" theoretical analysis.
Lemma D.18. Define the distributions P1 and P2 as follows:
Pi obtained by uniformly drawing E (0.1.....4 1) and then sampling uniformly at random from Σ
P2 obtained by uniformly drawing & (0, ,..,,,h. k- 1) and then sampling continuation to ho of length f, according to M
Let Ho = H {ho}. Assume that for some parameters a, 7
Esmp, (f)) [1
(5)
3 yuP, (Ho, (f {ft) + +
Then for any vector such that
E ENP, [[ -f
AA.
(6)
if we define
PLinGen is the distribution obtained by sampling from LinGen(t; v, H. k)
Pinuhh is the distribution of length k continuations of be according to M
then
D DKL(Pmen (PinGen) S 2k. aA + 71 )
Proof. Let u be the n 1-dimensional vector u == v1, Taking (5) and taking the inner product of both sides with un and then applying (6), we have
Eywp, (fhhol (ft) - TT CM, (M)) s as + 7(1 + [|0113)
(7)
Now assume that LinGen has sampled a sequence f = 21 ... 24 so far. Then by the definition of LinGen we can bound
DRL.
Prwv-lhho fl
PrlinGen(e,
< max log Pr (2]h0 o - log Pr PILingen(o.HLA)
< 2 max Las (=///0 S (iinGen(u.H,,)(22
(8)
< 22 {f}) T (MHHH((1)
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where the arguments on the LHS are distributions over the token space E given by the next-token probabilities the two different sampling procedures. Now using the KL chain rule, we get
DKL (Piutal PPLiGen) p~P
DKL Praslihoo
of Pr
2 ab + (1
where the last step follows by combining (7), (8) and Cauchy-Schwarz. This completes the proof.
Remark (Improving the Bound by Factor of [Σ]. The above proof can be "lossy", costing an extra factor of VEI when translating from KL L distance in (8). This factor can be saved, provided that the errors in the logits are roughly uniformly distributed over the token space this what we observe in practice.
We can also flip the arguments of the KL divergence match the quantity we measured empirically in Section 3.3, provided that the logits of the true language model are bounded.
Corollary D.19. Assume that LL [=]h]| S for any history h E E* and token z E. Then under the same conditions as Lemma 1.18, we also have
DKL. D (PLinGen PPnak)) S 2(1 + k(log Σ + 2C))VK (as + >(1 +
[|u1|()) 0.25
Proof. Note that the assumption implies
log [rM[2|M] S log | + 2C
for any h E E E. Now we can simply combine Fact D.21 and Lemma D.18 to get
DEL. (Pimn)) S (1 + k(log]
Σ + 2C)) / 2K inth PPnnGen
< 21 + A(log Σ vc (as + 711 + [10)13)) 0.25
as desired.
Finally, we also state concentration bounds for being able test (5) and (6) from samples.
Fact D.20. Let H C Σ' with [H]= n and let P be any distribution over sequences in Σ* Assume that the language model M satisfies the bound [L [h]] < C. for any history ht E Σ*. Then with 1 - 8 probability, if we let F be subset of S E C log(1/8) samples drawn from P, then
FF (H,
Esnp [Car (M (f3) Se
and consequently, for any v E R..
100 LM (M, FIl2 ENN LM, [/)))1 se||ul1
Proof. For the first inequality, note that
FEF
Also note M, (f))l first inequality. The second inequality follows directly from the first by taking the inner product of the LHS with T.
CV by assumption, Matrix Bernstein immediately gives the
Remark 4. As before, expect much faster concentration in practice e.g. used worst case bound on the individual summands when we applied matrix Bernstein, but a much tighter bound, that saves the dependence on [E], seems hold empirically because the columns of Lss (H, {f)) are generally not too correlated with each other.
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Fact D.21. Let P.Q be two discrete probability distributions elements given by P
(pr.....PP. Ps) and = {q1,.... :.......s..... 9s). Assume that -log 2 C for all i [s). Then
= (1 (
Proof. We can write
(Q]1)=
Σ
(log q log pt)
vE[o]
Σ
(: q: - -logpt
Σ
(a:
pllg + log
$ (1
C)
1 Pil
EE
(1+ (1+
Note that for the second inequality used the convexity of x log x and the final step follows from Pinsker's inequality.
E FUTURE DIRECTIONS
Our work motivates numerous directions for future work:
Can we better understand how the singular value decay evolves during training (recall Fig- ure 3) and can we use it as diagnostic for training progress?
The low rank structure allows us to represent each history as a vector (recall Section 1.1) can we extract concepts and features this representation space? This would help build towards model-agnostic approach to interpretability.
Can we use LINGEN or some variation of it to bypass safety guardrails and generate re- sponses unsafe prompts? While, as stated, LINGEN requires querying the model on hearg o f for many different choices of f, there are avenues for circumventing the need to reveal heargg For example, we can split up heagg = hą h2, treat hı the target history, and then ht2 as the first part of the generated future.
Related to the previous question, does our framework suggest techniques for safeguarding against such attacks?
Can extend theoretical results to the case when the model is only approximately low- rank as observed in practice? Further, it is an interesting open question to understand what notions of approximation (e.g. total variation vs matrix norm) are theoretically feasible while maintaining fidelity to practice.
F STATEMENT ON USE OF LLMS
We used LLMs to help us write and refine certain fragments of code e.g. we would provide a description of a function we wanted and ask an LLM help implement it. We also used LLMs to help us find related work and references and to help polish the writing after we had a draft.
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Reviews:
[{"review_id": "SwUm2nPspu", "reviewer": "Reviewer_7Jjj", "summary": "This paper proposes to analyze the low-dimensionality of LLMs' generation process by studying a curious construction called the extended logit matrix, which consists of the log-probability of the topK predicted tokens $\\log P(Z|H\\circ F)$, where $H$ and $F$ are segments of text randomly sampled from a certain source. They found low-dimensional behaviour in the resulting logit matrix, which suggests linear dependence - the predicted logits to some prompts could be closely approximated as an linear combination of predicted logits of other, statistically unrelated prompt. To prove this point, they introduce LINGEN, a linear generation algorithm that reconstructs the output distribution for a target prompt by linearly combining logits from unrelated prompts via a weighting vector $v$ obtained via linear regression. Surprisingly, this produces coherent text—even when the basis prompts are nonsense—revealing strong linear constraints in LLM's autoregressive generation.", "questions": "- Why is $v$ fit in log-probability space rather than pre-softmax logits, and how sensitive are results to this choice?\n- Could the observed low-rank structure be derived from, or correspond to, linear relationships in the model’s internal hidden states (e.g., out embedding)?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The framework is architecture agnostic: Studies LLMs through their output distribution geometry rather than internal activations.\n\n- The analysis of extended logit matrix is relatively straightforward and tractable via matrix decomposition algorithms.\n    \n- The fact that one could infer LLM generation of a target prompt using unrelated or even non-sensical prompts is very surprising.", "weaknesses": "- While the low-rank property is convincingly demonstrated, the paper stops short of providing a theoretical account for why transformer models should exhibit such linearity in distribution space. However, I do not hold this against the paper since the paper is already phenomenologically rich and self-contained.\n- The explanation of how the extended logit matrix is constructed, as well as its motivation, is quite abstruse. I think the paper could benefit from a cartoon illustration of the structure of the extended logit matrix, as well as the LINGEN algorithm. \n- Although LINGEN reproduces plausible text, the qualitative examples remain small-scale, and quantitative evaluations of long-horizon coherence or perplexity are limited.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Zp4uj81OW2", "reviewer": "Reviewer_PfEp", "summary": "The authors present an analysis of low-rank structures for logit matrices in language models. The paper idea is very interesting and new, since previous analyses limit to analyze logit structure for a prompt, here the authors consider logit matrices over sets of histories and futures. The experimental analysis reveals interesting aspects of low-eank structures, which are of extreme interest for the community working in interpreting language models internals.", "questions": "**Note that:** There is much intersection with works that have treated embeddings and unembeddings in language models. From [1,2], the logit matrix which is used in this paper has a peculiar relation to embeddings and unembeddings. By defining the mean-unembedding vector $\\psi_m := \\sum_{y \\in \\Sigma} \\psi(y) / |\\Sigma|$, then (if I'm not wrong) the logit matrix can be written as:\n$$\n{L_M}(\\mathcal H, \\mathcal F)_{(h, (f,z))} = \\phi(h \\circ f)^\\top (\\psi(z) - \\psi_m)\n$$\nwhich is again equal to considering the pivoting step in [2] and similar to as it is done in [3]. Interestingly, this casts a connection to theoretical results in linear properties of language models [1,2] for next-token predictors. Low-rank is new in this sense, but there can be some interesting connection to what are parallel vectors and parallel histories (as you notice in your paper), but also to the effective complexity of the language model [2].  \n\n**Q** For the versions of OLMO, what is the representation dimensionality $d$? Did you check if using $d$ components you recover the distribution of original LLM? \\\nI think that drawing a connection to effective complexity/dimensionality of representations can highlight distinction from the setup in [1,2] which is not concerned with embeddings of futures. This may inspire an update of effective complexity of the model based on histories and futures.\n\n**Q** About results in Sec 3.2, what is the difference in evaluating the principal angles between rank reduced A and A' w.r.t. ones? For rank reduction of A and A' with rank k, if you keep the k-th higher components, should that be the same if you consider rank 2k? I don't understand how this measure depends on rank reduction. Can you further explain?\n\n**Q** What is the authors opinion about the relation to linear properties that are observed for single-token logits? For example, the ISAN construction much resembles relational linearity in a sense. From [2,4,5], the mapping from context to tokens can be seen as relational linear if you have that $z = A_f \\phi(h)$, for some specific matrix $A_f$ that only depends on the future $f$. It seems the ISAN model implies something similar for all members of the chain. \n\n------\n\n[3] Hinton et al. \"Distilling the knowledge in a neural network\" (2015) \\\n[4] Hernandez et al. \"Linearity of relation decoding in transformer language models\" ICLR 2024 \\\n[5] Paccanaro and Hinton \"Learning Distributed Representations of Concepts using Linear Relational Embedding\" 2020", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "I found that the experimental section is almost excellent and appreciate how the authors give a comprehension of the phenomena observed, state new hypothesis and propose future directions. On its own, this is a great contribution which opens the doors for more future research. \n\nAbout experiments:\n1.  While measuring the complete logit matrix is unfeasible for all tokens and only top-100 are considered, the analysis clearly shows low rank structures across different datasets with histories and futures size $n \\in (10^3,10^4)$, both for a logit matrix  $n \\times n \\times k$ for full set of histories and futures, and also for downsized $n/t \\times n/t \\times k$, varying $t$ (correct me if I got it wrong).  The scaling of eigenvalues is an interesting observation, which is well explained by the authors through the analysis of  the coefficient $\\alpha$. \n2. The experimental outcome of principal angles between logit matrix of sensical  (A) and non-sensical futures (A') is also an interesting observation. As explained by the authors, this shows that despite low-rank approximation of A and A',  the two spaces have high intersection, with not much beyond the intersection. The finding that this also holds between different language models is even more interesting, it should be highlighted more in the paper. \n3. Using these linear dependencies between histories to control generation with LINGEN is an interesting application and a nice methodological contribution. The only limitation is that LINGEN requires having access to (many) logits of the target history for many futures to evaluate the corresponding $v_{h_{tar}}$. Still, the method is interesting and the authors can highlight more how to improve such limitation in evaluating $v_{h_{tar}}$ (even in a limitation section). In my opinion, a careful choice of futures could help in that direction.\n\nAbout theory:\n* The analysis on time-varying ISAN is valuable, with novel theoretical insights, although with very limited explanation in the text. I am particularly interested in how such a linear model can capture generation in a generic language model, see my questions below.", "weaknesses": "While the material is excellent, with new interesting results and a theoretical reformulation of the ISAN model, the clarity should be improved. \nMy comment arise from the fact that I had to re-read the text many times and continue jumping to appendix before completely capturing the message or familiarizing with the quantities the authors introduce. In fact, while theoretical explanations are useful in the experimental sections, it overloads the reader trying to grasp both the theory and interpret the experiments at the same time. \nI suggest the authors to move, as much as possible, theoretical matherial in an apposite section while leaving more space to experimental details alone. \n\nOther points that are weak:\n1. **Notation** Denoting with $\\mathcal H$ and $\\Sigma$ the sets of histories and of tokens, it is a bit imprecise to write $\\mathbb R^{\\mathcal H}$ and \n$\\mathbb R^{\\Sigma}$ the vector spaces with dimension of the sets. It would be more convenient to use the cardinality or introduce naturals $M$ and $N$ for that. To be clear, this was a bit confusing when reading the manuscript. \n2. **Notation** From other papers, feature mappings $\\phi, \\psi$ are also called the embeddings and unembeddings of the model [1,2]. \nI also noticed that while $\\phi: \\mathcal H \\to \\mathbb R^d$, with $d$ dimension of vector embeddings, $\\psi: \\Sigma \\to \\mathbb R^d$. \nSo it is a bit imprecise to write $\\psi(f)$, if $f \\in \\mathcal F$  is a sentence. Can the authors explain the relation to formalism in [1,2]? \\\nI noticed that the authors make use of $\\phi(h \\circ f)$ in Def. 2.2, which makes sense, but suggest that $\\psi(f)$ is never used in any step of the computations.\n3. **Experiments** Some quantitative results should be made more clear. In 3.2, what is the proportion of \"small\" principal angles of the two ranks approximations? And how this impacts to finding transfer across different futures? The claim \"Thus, linear relationships between histories transfer to significantly different sets of futures.\" should be substantiated more. In light of this precisation, it would be beneficial to highlight more the results on different language models, which essentially test a different hypothesis (which tells that the vector $v$ of the null space is almost shared between models, if I got it correctly). This aspect should be explained further. \n4. **Notation** In lines 359/360 I guess there is a typo in saying $t\\geq 1$, since you would have $z_{1:0}$ for $t=1$. \nAlso denoting with $v$ the vector associated to $h_{tar}$ is a bit confusing. I suggest to denote it as $v_{tar} \\in \\mathbb R^{|\\mathcal H|}.$ \n5. **Method** LINGEN is interesting, but as far as I can see, estimatingthe vector $v$ requires prompting many times the language model with the history $h_{tar}$. Given the scope of the paper, this is not a serious limitation, but the discussion should be enlarged on this. I expect that some tricks can be created to estimate the vector $v$ for $h_{tar}$ without prompting all the futures. What is the authors' opinion on this matter?\n6. The connection to ISAN is interesting but received far too limited space. I could not toatally grasp the message in that section, especially for why a time-varying ISAN would be more preferable than a language model with low-rank. My doubt arises as you need to instantiate $A_{z, t}$ and $B_t$ for each element $t$ in the chain, and each token $z$. Withouth sharing $A$ and $B$ across elements $t$ of the chain, what is the advantage of this construction?  Can you further elaborate? \\\nThere is also a lot of material on ISAN which would constitute a separate contribution and that I could not review from the Appendix. Because of the lenght of the material in the Appendix and the statements not appearing in the main text, I could not verify the correctedness of the claims. I really suggest the authors to take more space to explain the construction and relevant results of ISAN.\n\n**Clarity comment** Overall, some results could be moved to Appendix or shortned. E.g., the whole introduction of KL in Def. 3.1 and discussion can be easily moved to the Appendix without compromising much the text, as they are quite standard,  while leaving space for expanding on some points I mentioned above. \n\n**Comment** Improving clarity would much increase the impact of the authors' contribution, which is extremely timely and relevant for research in language models structure learning. Depending on the rebuttal, I am available to further review a restructuring of the paper with my and other reviewers comments. \n\n-------\n\n[1] Park et al. \"The linear representation hypothesis and the geometry of large language models\" ICML 2024 \\\n[2] Marconato et al.  \"All or none: Identifiable linear properties of next-token predictors in language modeling\" AISTATS 2025", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "s80Ps3za4i", "reviewer": "Reviewer_L59i", "summary": "This paper proposes a novel, model-agnostic framework for understanding the internal structure of language models by studying their \"extended logit matrices.\" The authors define this matrix based on the log probabilities of sequences (futures) given various contexts (histories).\n\nThe paper's contributions are twofold:\n\n- Empirical: They demonstrate across a range of modern LLMs that these extended logit matrices have a consistent, approximate low-rank structure. This structure is shown to emerge early in the pre-training process. This low-rank property is then exploited to create a generation procedure \"LINGEN\"， which can generate a response to a target prompt by querying the model on unrelated or even nonsensical prompts.\n\n- Theoretical: They connect this empirical finding to a formal generative model, the Input Switched Affine Network (ISAN). They prove that having an exact low logit rank is equivalent to being expressible as a time-varying ISAN and provide provable learning guarantees for this model class under logit query access.", "questions": "- Following up on the weakness: Could you elaborate on the practical feasibility of using LINGEN to bypass existing safety alignments? How many histories would be required? Is it feasible to perform the attack on large LLMs? How much cost would it take to construct these histories and their corresponding weights for linear combination?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "- **Novel Framework**: The concept of the \"extended logit matrix\" is a simple and powerful model-agnostic abstraction. It provides a new lens to study the intrinsic dimensionality and structure of LLMs beyond just analyzing weights or single-token logits.\n\n- **Comprehensive Validation**: The paper does an excellent job supporting its claims with both strong empirical evidence and solid theoretical grounding. The empirical study is thorough, covering multiple model architectures and sizes, and the theoretical connection to ISANs is elegant.\n\n- **Novel Generation & Safety Implications**: The \"LINGEN\" generation method is a surprising consequence of the observed low-rank structure. The ability to generate text from a target prompt using only queries to unrelated, nonsensical prompts is a significant finding. This has clear and important implications for AI safety, as it suggests a potential new vector for bypassing prompt-based safety filters.", "weaknesses": "- **Safety Implications Discussed but Not Fully Explored**: The primary weakness is that the safety implications, while highlighted, are not demonstrated in practice. The paper suggests that LINGEN could be used to \"circumvent input filters,\" but it does not provide a concrete experiment showing such a jailbreak. The analysis is currently a proof-of-concept (i.e., generating coherent text) rather than a practical demonstration of a safety bypass. This is acknowledged as future work (Appendix E), but it leaves the practical severity of this potential vulnerability unclear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Oral)