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REVIEW 4 major objections 5 minor 2 cited by

LOST: Low-rank and Sparse Pre-training for Large Language Models

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Low-rank weights plus a frozen sparse channel mask can match full-rank LLM pretraining.

desk verdict A credible incremental method for low-rank plus sparse LLM pre-training with solid ablations, but the headline gains over full-rank rest on inherited baselines and single-seed runs. read the letter →

arxiv 2508.02668 v1 pith:DH2MY2Z4 submitted 2025-08-04 cs.LG

Jiaxi Li , Lu Yin , Li Shen , Jinjin Xu , Liwu Xu , Tianjin Huang , Wenwu Wang , Shiwei Liu
show 1 more author
Xilu Wang
This is my paper · ORCID
classification cs.LG
keywords LLMpretraininglow-ranktrainingsparseplusdecompositionSVDinitializationchannel-wisesparsitymemory-efficientLLaMA
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to show that large language models can be pre-trained from scratch without ever storing or updating a full-rank weight matrix: replace each weight with a low-rank factor plus a channel-wise sparse residual, choose both pieces from a single SVD of the initial matrix, and let gradient descent update them. Tested on LLaMA-style models from 60M to 1B on C4, LOST reports lower or equal perplexity than full-rank training at every scale except the 350M point, where it is close, while cutting memory nearly in half at 1B (3.66 GB vs 8.04 GB). A 7B run over 40K steps also reports lower perplexity than standard 8-bit Adam, and an 8-bit variant of the method halves memory again. If the claim holds, low-rank-plus-sparse weight decompositions are a viable substitute for full-rank weights during pre-training under tight memory budgets.

What carries the argument

The load-bearing mechanism is a one-time SVD-based co-design of the two components. The low-rank factor is initialized from the top-$r$ singular vectors, so its product approximates the original weight in Frobenius norm; the sparse residual is initialized from the remaining singular subspace, so it carries directions orthogonal to the low-rank part instead of duplicating it. Channel-wise structure, keeping whole columns rather than element-wise entries, makes the store compact because only $k$ column indices need to be saved. The activation inserted between $A$ and $B$ adds nonlinearity without adding parameters. Together this preserves both the dominant subspace and a small set of high-energy output channels, and the frozen mask means the SVD cost is paid only once, before training.

What would settle it

Train a 130M LOST model but re-run the SVD and channel selection every 5,000 steps using the current factored weights; if perplexity drops below the reported 24.05, the fixed mask is the load-bearing cost. A cheaper check is to replace the SVD-selected channel set with randomly chosen channels at initialization and compare perplexity: if random channels match, the complementary-spectrum selection is not the cause.

Watch

Extended reading notes

Core claim

LOST's central discovery is that the information lost by truncating a randomly initialized weight matrix to rank $r$ does not need to be modeled densely: it can be captured by keeping $k$ whole output channels selected from the complementary subspace. Concretely, a single SVD of the Kaiming-initialized matrix $W$ gives $W_l = AB^T$ with $A = U_r \Sigma_r^{1/2}$, $B = V_r \Sigma_r^{1/2}$, with a SiLU activation between the two factors; the residual subspace $W_{\text{comp}} = \sum_{i>r} \sigma_i u_i v_i^T$ is used to score each channel by its L2 norm, and the top $k = \lceil \rho n \rceil$ channels are kept as $W_s = W[:, I]$. The forward pass is $o = \gamma\, \sigma(xA)B^T + (1-\gamma)\, x[:,I] W_s^T$. With $\gamma=0.7$, $\rho=0.01$, and rank adjusted so the parameter count matches other low-rank baselines, LOST obtains 32.25 vs 34.06 perplexity at 60M, 24.05 vs 24.36 at 130M, 18.95 vs 18.80 at 350M, and 15.02 vs 15.56 at 1B, with estimated memory of 0.24, 0.57, 1.11, and 3.66 GB versus 0.35, 0.81, 2.21, and 8.04 GB for full-rank training.

Load-bearing premise

The sparse channel mask is fixed after initialization, so the whole method assumes that the channels that look important before training stay important all the way through.

Editorial extensions

If this is right

  • With the same token budget, LOST reports lower validation perplexity than full-rank training at 60M, 130M, and 1B, and effectively matches it at 350M.
  • The parameter count for a linear layer drops from $mn$ to $r(m+n) + mk$; at 1B this yields 3.66 GB estimated memory instead of 8.04 GB.
  • At 7B over 40K steps, LOST reports perplexity 16.48 versus 18.09 for 8-bit Adam, and 8-bit LOST uses 50.19 GB versus 72.59 GB.
  • Fine-tuning RoBERTa-base on GLUE with $r=7$ gives an average score of 86.51, above the full-size model's 86.28 in the paper's table, suggesting the decomposition transfers beyond pretraining.
  • Ablations show that selecting channels from the residual singular subspace outperforms alternatives derived from the largest, smallest, or random singular values.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the channel mask is frozen after initialization, the method implicitly assumes the SVD spectrum of the initial weight predicts which channels stay important through training; an adaptive re-selection schedule is a natural test, and if it helps, the fixed mask is a bottleneck rather than a feature.
  • The channel-wise structure may be a stronger design choice than the paper fully exploits: whole stored columns could combine directly with hardware-friendly structured sparsity kernels at inference time, something element-wise sparse methods cannot do without conversion.
  • The authors flag that models above 7B remain untested; if the one-time SVD split reproduces at 10B-plus scale, it would make from-scratch pretraining feasible on far smaller hardware, while longer training schedules would stress-test the frozen-mask assumption.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes LOST, a method for pre-training LLMs from scratch using a sum of a low-rank component and a channel-wise sparse component. The low-rank factors are initialized from the top-r singular vectors/values of the initialized full-rank weight matrix, and the sparse channels are selected, once at initialization, from the residual subspace using an L2-norm importance score; the two branches are combined at the output level with a fixed trade-off coefficient gamma. The authors evaluate LOST on LLaMA-style models from 60M to 7B parameters trained on C4, reporting perplexity, parameter counts, and memory estimates, and they also test a fine-tuning variant on GLUE. The central claim is that LOST reaches lower or comparable validation perplexity than full-rank training while using fewer parameters and less memory.

Significance. If the central claim holds, LOST would be a practical contribution: it provides a parameter- and memory-efficient way to pre-train transformer LMs without a full-rank warm-up, with a clean SVD-based initialization that is performed once. The paper includes a thorough ablation study on the complementary matrix source, channel selection criterion, initialization scheme, activation placement, sparsity level, and trade-off coefficient, and it reports experiments up to 7B parameters plus fine-tuning results on GLUE. The method is clearly described and the code is promised to be released. However, the headline performance claim rests on a comparison whose full-rank and several baseline numbers are not reproduced in the same codebase, and no seeds or error bars are given; the margins at 130M and 1B are small enough that the central claim is not yet established by the evidence presented.

major comments (4)
  1. [Section 4.1, Table 1] The headline comparison against full-rank training is not a matched comparison. The text states that only LORO and CoLA results are reproduced with their default scripts, while Full-Rank, LoRA, ReLoRA, GaLore, and SLTrain numbers are directly reported from [16, 20]. The claimed advantages over full-rank are 1.81 PPL at 60M, 0.31 PPL at 130M, and 0.54 PPL at 1B; at 350M LOST is actually 0.15 PPL worse than full-rank. These margins are comparable to typical cross-codebase differences in tokenizer, batch size, learning-rate schedule, and numerical format. To support the claim 'LOST achieves comparable or superior performance compared to full-rank models,' all baselines, especially full-rank, must be rerun in the same codebase and with the same training setup, or the comparison must be explicitly downgraded to a cross-paper reference.
  2. [Section 4.1 / Experimental setup (no seeds)] No random seeds, replicates, or error bars are reported anywhere in Section 4. The main PPL differences over full-rank are small in absolute terms (0.31 at 130M, 0.54 at 1B), and without variance estimates it is impossible to know whether LOST's advantage is statistically meaningful. The authors should run the key comparisons (at least the 60M, 130M, and 1B models) with multiple seeds and report the mean and standard deviation, or at least the best-of-three with the spread.
  3. [Section 4.1, Table 2 and Appendix Table 11] The 7B scaling claim is based on only 40K training steps (with LOST extended to 150K), while the paper's own Table 13 indicates a full epoch would require roughly 19.7B tokens. Moreover, in Table 11 the Full-Rank Adam baseline is run with batch size 4, whereas LOST runs with batch size 8, so the reported memory figures per GPU (49.53 GB vs 62.15 GB) are not comparable, and the full-rank run's instability after 12K steps is observed on a different batch size and a truncated schedule. A 40K-step comparison does not establish that LOST trains stably to convergence at 7B relative to full-rank. Please either extend the full-rank baseline to the same number of steps and batch size, or soften the scalability conclusion.
  4. [Section 3.2 / Algorithm 1, Step 2] The sparse channel mask I is selected once from the SVD of the randomly initialized weight matrix and is never updated during training. The method therefore assumes that the channels that are most important at initialization remain the channels that are most important throughout training. The ablation in Table 3 shows that the specific choice of the complementary matrix changes PPL by only about 0.1–0.2 (e.g., 32.25 vs 32.35 at 60M), which undercuts the paper's emphasis on the 'complementary co-design' as the source of the improvement. The mechanism claim (contribution 2) would be strengthened by an experiment tracking how much the selected channels drift during training and whether an adaptive mask re-selection would improve performance.
minor comments (5)
  1. [Section 3.1, Eq. (1)] The dimensions are inconsistent: the text defines W ∈ R^{m×n}, but Eq. (1) writes W ∈ R^{n×n} and the singular value matrix diag(σ_1,...,σ_n) ∈ R^{m×n}; please make the rectangular case explicit.
  2. [Table 3] The column header SVDrand_l2 appears twice; one of the entries is presumably intended to be a different variant (for example SVDrand_l1).
  3. [Section 2.2] The phrase 'low-rank LLM pertaining' should be 'pre-training'.
  4. [Section 4.1 / Table 11 caption] The caption for Table 11 says 'for 40K steps' but the table reports results at 10K, 40K, 80K, 120K, and 150K; the caption should reflect the full step range.
  5. [Appendix Table 12 and text] The comparison between structured and unstructured sparsity is not completely fair because the unstructured variant has to be processed separately to accommodate the activation between low-rank factors; this extra requirement is described in the text, but the memory comparison would be clearer if the activation-free structured case were also reported.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: LOST's SVD-based low-rank/sparse construction is empirical and self-contained; only non-load-bearing self-citations.

full rationale

The central derivation is an empirical training recipe, not a quantity reconstructed from fitted inputs. SVD is applied once to the random initial weight matrix W (Eq. 1); the low-rank factors A,B and the channel mask I are fixed from the untrained spectrum, and the reported perplexities are measured after gradient-descent training of A,B,Ws. The claim that LOST 'achieves competitive or superior performance compared to full-rank models' is therefore a testable experimental outcome, not a consequence of defining the method in terms of the target metric. Baselines are mostly inherited from external papers [16,20], with LORO and CoLA reproduced locally; none of these comparisons is a self-citation chain. The authors do cite their own prior work (e.g., Q-GaLore [19], WeLore [13], ICASSP [15]), but those citations are contextual and not used to justify the core claim. The choice of gamma=0.7, sparsity=0.01, and rank 256 is tuned on C4 validation ablations; that is a standard hyperparameter-selection risk rather than a circular step, and Table 9 shows the result is robust across gamma in [0.6,0.8]. A separate technical concern is that Eq. (3) defines Ws = W[:,I] rather than Wcomp[:,I], so the sparse term is not literally the residual subspace claimed in Section 3.2; this is an implementation/claim mismatch, not circularity, and it does not make any reported number equal to an input by construction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central result rests on standard ML assumptions and tuned hyperparameters, not on new physical entities or new mathematical axioms. The two most consequential entries are the stationarity of the SVD-based low-rank subspace and the fixed channel mask, and the comparability of inherited baseline numbers.

free parameters (4)
  • gamma = 0.7
    Output-level trade-off between low-rank and sparse components; chosen from ablation on the same C4 validation (Table 9).
  • sparsity ratio rho = 0.01
    Fraction of channels retained in the sparse component; selected from the fixed-budget ablation (Table 8) and kept fixed across all model sizes.
  • Wcomp rank = 256
    Number of singular values used to build the residual matrix for channel selection; ablated in Table 7 and fixed globally.
  • per-model low-rank rank = not stated in text; lower than nominal r to match LORO/CoLA parameter counts
    Table 1 caption says LOST uses an actual rank lower than r, but exact ranks are not reported, making parameter-count claims hard to verify independently.
assumptions (5)
  • domain assumption SVD of the randomly initialized weight matrix identifies a low-rank subspace that remains useful after gradient updates.
    Introduced in Section 3.1, Eq. 1; the method never re-computes SVD during training, so this is a load-bearing premise.
  • domain assumption The L2 norm of residual channels from Wcomp is a valid importance score for choosing which channels to keep.
    Eq. 2; validated only by ablation (Table 3), with no theoretical or statistical justification.
  • domain assumption A fixed set of channels I selected before training continues to be the right sparse support throughout training.
    Algorithm 1 Step 2 selects I once; training updates Ws but not the mask, so the support is assumed stationary.
  • domain assumption Perplexity on C4 validation after a single epoch is a reliable proxy for LLM pretraining quality.
    Standard practice in the cited baselines, but it is an assumption about evaluation.
  • domain assumption Baseline results from prior papers are comparable even though not rerun in the same harness.
    Table 1 caption states results for other methods are directly reported from prior work; this assumes identical setups.

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Cite this review

Pith. "Pith review of LOST: Low-rank and Sparse Pre-training for Large Language Models." pith.science (2026). https://pith.science/paper/DH2MY2Z4

@misc{pith2026250802668,
  author       = {Pith},
  title        = {Pith review of: LOST: Low-rank and Sparse Pre-training for Large Language Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DH2MY2Z4}},
  note         = {Machine review of arXiv:2508.02668}
}
read the original abstract

While large language models (LLMs) have achieved remarkable performance across a wide range of tasks, their massive scale incurs prohibitive computational and memory costs for pre-training from scratch. Recent studies have investigated the use of low-rank parameterization as a means of reducing model size and training cost. In this context, sparsity is often employed as a complementary technique to recover important information lost in low-rank compression by capturing salient features in the residual space. However, existing approaches typically combine low-rank and sparse components in a simplistic or ad hoc manner, often resulting in undesirable performance degradation compared to full-rank training. In this paper, we propose \textbf{LO}w-rank and \textbf{S}parse pre-\textbf{T}raining (\textbf{LOST}) for LLMs, a novel method that ingeniously integrates low-rank and sparse structures to enable effective training of LLMs from scratch under strict efficiency constraints. LOST applies singular value decomposition to weight matrices, preserving the dominant low-rank components, while allocating the remaining singular values to construct channel-wise sparse components to complement the expressiveness of low-rank training. We evaluate LOST on LLM pretraining ranging from 60M to 7B parameters. Our experiments show that LOST achieves competitive or superior performance compared to full-rank models, while significantly reducing both memory and compute overhead. Moreover, Code is available at \href{https://github.com/JiaxiLi1/LOST-Low-rank-and-Sparse-Training-for-Large-Language-Models}{LOST Repo}

Figures

Figures reproduced from arXiv: 2508.02668 by the authors.

Figure 1
Figure 1. Performance comparison of pretraining methods on LLaMA-1B (C4 dataset). Smaller cir￾cles in the lower-left indicate better memory effi￾ciency and lower perplexity. See [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Framework of the LOST method. LOST begins by initializing a full-rank weight matrix [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Breakdown of memory consumption across different methods on 1B model. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. SLORR: Simple and Efficient In-Training Low-Rank Regularization

    cs.LG 2026-07 accept novelty 6.0 of 10

    A stateless, SVD-free regularizer approximates polar factors to induce low-rank weight structure during training, enabling better post-training compression of vision models and LLMs at under 8% overhead.

  2. SALAAD: Sparse And Low-Rank Adaptation via ADMM for Large Language Model Inference

    cs.LG 2026-02 conditional novelty 6.0 of 10

    SALAAD uses an ADMM-style optimizer and an adaptive controller to force LLM weights into sparse-plus-low-rank form during pretraining, so one checkpoint can be elastically deployed at many parameter budgets.

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.