D-SLR: The Disjoint Row-Sparse plus Low-Rank Decomposition

arXiv:2610.10636 · cs.LG, stat.ME, stat.ML · Submitted 2026-10-07 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "D-SLR: The Disjoint Row-Sparse plus Low-Rank Decomposition".

Jane: The gist The Disjoint Row-Sparse plus Low-Rank Decomposition proposes a closed-form drop-in for the truncated SVD that improves or exactly matches it by restricting rows to either being stored verbatim…

Tom: First, who's behind it and why it matters.

Paper summary: Tom: So we're looking at this paper called D-SLR: The Disjoint Row-Sparse plus Low-Rank Decomposition. What's the big idea here?

Jane: Basically, it proposes a new way to compress a matrix for reconstruction that improves or exactly matches what we get from the truncated SVD, which is usually just using a single low-rank structure.

Tom: The authors say they restrict rows to either being stored exactly or being approximated by that low-rank fit, but never both. They claim this restriction costs nothing under squared error because you can find the best solution disjointly with fewer parameters at every rank and row count shape.

Lu: That sounds like they are solving a joint problem involving row sparsity and low-rank approximation, which is usually really hard because it's non-convex and combinatorial.

Meng: So they're trying to tackle that difficulty by making the choice explicit—store this row or approximate it with the low-rank structure.

Lalam: And under squared error, this restriction costs nothing, as the joint optimum is attainable disjointly with fewer parameters at every nontrivial rank and stored row count shape.

Tom: Right, so if you have zero stored rows, D-SLR just becomes the truncated SVD, which means it never does worse at equal cost.

Jane: But they don't stop there; they actually build a whole scoring system that shows the error versus parameters tradeoff and then you pick what you want based on an error target or parameter count.

Tom: That sounds like a pretty complete package for choosing how much to compress, even when dealing with row sparsity.

Jane: Now let's look at the conclusion of this paper, D-SLR: The Disjoint Row-Sparse plus Low-Rank Decomposition. What does it actually mean for us?

Tom: It seems like the main point is that D-SLR is a drop-in for the truncated SVD because it never does worse when you spend the same number of parameters on both methods.

Jane: And they provide an assumption-free certificate from just W alone that lower bounds the error for every row sparse solution, and they show how you can tighten that floor by applying it to blocks of rows.

Lu: That certificate is computed after the fit for a given matrix and assumes no model of the data, which is pretty powerful because it doesn't rely on any prior knowledge about what the data looks like.

Meng: And they also provide a closed-form algorithm with no tuning, scoring every rank and row count with an upper bound on the error at the cost of three SVDs and no iterations.

Lalam: That means you can check if a rival solution could have a lower score by comparing its score Q(y') with the incumbent's score q-hat.

Tom: So, overall, it's this method that scores every shape with a closed-form upper bound on the error at the cost of three SVDs and no iterations.

Jane: It helps most on tall matrices where rows fit the shared basis poorly, and otherwise it just returns the truncated SVD.

Tom: So, to wrap up, D-SLR is this drop-in for the truncated SVD that never does worse at equal cost, costs three SVDs, and tells you how far off you could be from the best.

Jane: It can also be combined with further compression such as quantization.

Tom: That means it's a solid tool for finding a good balance between storing rows explicitly and using the low-rank structure.

Conclusion: Tom: So we're wrapping up D-SLR, the Disjoint Row-Sparse plus Low-Rank Decomposition. Basically, this method offers a direct replacement for truncated SVD when you want to be careful about both row sparsity and low-rank approximation at the same time.

Jane: It’s a closed form drop-in for the truncated SVD that never gets worse when you spend the exact same number of parameters on both storage methods.

Tom: Right, it’s this idea that you either store a row exactly or approximate it with a low-rank fit, but not both simultaneously.

Lu: The core idea is that this choice is free under squared error because you can find the best solution separately for every combination of stored rows and rank.

Meng: So if I have a matrix and I'm trying to compress it, D-SLR gives me a clear path to choose between storing specific data or using that low-rank shortcut.

Jane: It also gives us this certificate derived only from the matrix itself that sets a lower bound on the error for any row sparse solution we try.

Tom: That means you can check if your current compression choice is actually the best possible one, and it does this without needing any extra assumptions about how the data was generated.

Lalam: This certificate acts like a guide telling us exactly how much error we could still expect by trying different ranks and row counts.

Jane: It’s a really clean way to score these trade-offs, showing you the error versus parameters for every possible shape of compression.

Tom: And the authors provide this whole scoring system using just three SVDs and no iterations, which is pretty efficient for figuring out what works best.

Lu: It really helps when we have tall matrices where some rows just don't fit well into a shared low-rank basis, otherwise it defaults back to the standard truncated SVD.

Meng: So for practical AI applications, this means we can get a better estimate of how much performance we lose by choosing one compression style over the other.

Jane: It’s not just about getting the lowest error; it’s about having that certificate to prove you've found a good spot on the frontier.

Tom: Next up, we're looking at how this structure can be combined with other techniques like quantization to make it even more compact.

Vincent Szolnoky

Chalmers University of Technology

cs.LG, stat.ME, stat.ML

Submitted: 2026-10-07

Updated: 2026-10-07

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 92/100

The gist: The gist The Disjoint Row-Sparse plus Low-Rank Decomposition proposes a closed-form drop-in for the truncated SVD that improves or exactly matches it by restricting rows to either being stored

Key concepts

Disjointness is free: This means the optimal solution for problems involving both row sparsity and low-rank approximation can be found by choosing one approach exclusively: either store the row exactly or approximate it with a low-rank fit. The method shows that this disjoint choice never results in a worse outcome than trying to use both simultaneously, especially when considering parameter counts.
Certificate derived from W: The paper derives an assumption-free certificate based only on the matrix W. This certificate provides a lower bound on the error for every possible row-sparse solution. It can be refined by applying it to groups of rows and helps determine the potential benefit of choosing different ranks or stored row counts.
Universal Floor: This is a theoretical minimum error level that any solution must meet, regardless of how many parameters are used. This floor is calculated based on the maximum error achievable by combining a low-rank approximation with an exact fit for every possible shape. It helps define the best possible performance achievable within a certain parameter budget.

Terminology

Outlier Pursuit — 18× in this paper
Sparsity — 6× in this paper · explained in 4 episodes
Calibration — 3× in this paper · explained in 7 episodes
Regularization — 3× in this paper · explained in 3 episodes
Verification — 3× in this paper · explained in 2 episodes
Core — 2× in this paper · explained in 2 episodes
Fingerprinting — 1× in this paper · explained in 2 episodes
Matrix Completion — 1× in this paper · explained in 4 episodes
Pruning — 1× in this paper · explained in 4 episodes
Reconstruction Error — 1× in this paper · explained in 2 episodes

Summary

The gist The Disjoint Row-Sparse plus Low-Rank Decomposition proposes a closed-form drop-in for the truncated SVD that improves or exactly matches it by restricting rows to either being stored verbatim or approximated by the low-rank fit, never both<ref:2610.10636#pg6>

How it works

The D-SLR decomposition addresses the difficulty of jointly solving problems involving row sparsity and low-rank approximation, which is often non-convex and combinatorial<ref:2610.10636#pg5> The method restricts rows to either being stored verbatim or approximated by the low-rank fit, never both<ref:2610.10636#pg6> Under squared error this restriction costs nothing, as the joint optimum is attainable disjointly with fewer parameters at every nontrivial rank and stored row count (shape)<ref:2610.10636#pg6> With zero stored rows D-SLR reduces to the truncated SVD, so it never does worse at equal cost<ref:2610.10636#pg6> The algorithm scores the entire error-versus-parameters tradeoff, and the solution is chosen afterwards by a supplied error target or parameter count, or by a selection rule<ref:2610.10636#pg6>

Key Contributions

The paper makes three main contributions to this decomposition<ref:2610.10636#pg6>

Disjointness is free, meaning the optimum of the overlapping problem (Problem 1) is attainable by a disjoint solution that needs fewer parameters than the overlapping one whenever both rows and rank are used<ref:2610.10636#pg6> With no stored rows D-SLR is the truncated SVD, so at equal parameter count it is never worse than the SVD<ref:2610.10636#pg6>
An assumption-free certificate is derived from W alone, which lower-bounds the error of every rowsparse solution and can be tightened by applying it to blocks of rows<ref:2610.10636#pg6> This certificate provides a computable certificate on the potential gain of any other choice of rank and stored rows<ref:2610.10636#pg6>
A closed-form algorithm with no tuning is provided, scoring every rank and row count with a closed-form upper bound on the error, at the cost of three SVDs and no iterations<ref:2610.10636#pg6>

Theoretical Guarantees

The analysis establishes universal lower bounds for the error at every shape, which gives each solution a computable certificate on the potential gain of any other choice of rank and stored rows<ref:2610.10636#pg6> Corollary 4 defines the universal floor as E(X; r, k) = max T(X; r+k), T(X; r)−R(X; k), 0<ref:2610.10636#pg6> This floor bounds the frontier F(c) from below, meaning the lowest error reachable within c parameters is greater than or equal to the minimum E(r, k) for a given cost c<ref:2610.10636#pg6> Theorem 5 provides a score certificate that allows one to check if a rival solution could have a lower score by comparing its score Q(y') with the incumbent's score qˆ<ref:2610.10636#pg6>

Algorithm and Practicality

The D-SLR algorithm proceeds in four steps: Step 1 involves computing the SVD of W and flagging rows that fit a low-rank probe of W worst, followed by computing a second SVD of the unflagged rows only to find the basis Vˆ<ref:2610.10636#pg6> Step 2 is a sweep where every shape is scored with the surrogate error, and Step 3 involves picking a description using an EBIC-style price (Equation 8)<ref:2610.10636#pg6> Finally, Step 4 performs the exact fit by storing the chosen rows verbatim and fitting the rest with one truncated SVD<ref:2610.10636#pg6> The algorithm costs three SVDs regardless of the grid size, and no step iterates<ref:2610.10636#pg6>

Experimental Validation

Experiments on synthetic and real data (LLM embedding tables, network traffic, hyperspectral images) confirm the gains and quantify the certificate<ref:2610.10636#pg6> On LLM embedding tables, when retaining only half the table’s parameters, DSLR changed about ten times fewer of the model’s predictions than the truncated SVD<ref:2610.10636#pg6> D-SLR beat the truncated SVD on all three measures at every parameter count on all three matrices<ref:2610.10636#pg6> The split floor certified D-SLR’s solution far more tightly than the universal floor on the torture test<ref:2610.10636#pg6>

Conclusion

D-SLR is a drop-in for the truncated SVD that is never worse at equal cost, costs three SVDs, and states how far it could be from the best<ref:2610.10636#pg6> It helps most on tall matrices with rows that fit the shared basis poorly, and otherwise returns the truncated SVD<ref:2610.10636#pg6> The certificate prunes much like branch-and-bound (BnB), with each shape playing the role of a box<ref:2610.10636#pg6> The split floor on the other hand, of Section 6, closes most of these shapes<ref:2610.10636#pg6> This makes D-SLR a drop-in for the truncated SVD that is never worse at equal cost, costs three SVDs, and states how far it could be from the best<ref:2610.10636#pg6> It can also be combined with further compression such as quantisation<ref:2610.10636#pg6> The paper introduces D-SLR, which stores each row verbatim or fits it with a shared low-rank basis, never both<ref:2610.10636#pg6> It helps most on tall matrices with rows that fit the shared basis poorly, and otherwise returns the truncated SVD<ref:2610.10636#pg6> The default pick assumes the leftover error is noise, and its certificate loosens when many rows and ranks are affordable<ref:2610.10636#pg6> It can also be combined with further compression such as quantisation<ref:2610.10636#pg6> It helps most on tall matrices with rows that fit the shared basis poorly, and otherwise returns the truncated SVD<ref:2610.10636#pg6> The default pick assumes the leftover error is noise, and its certificate loosens when many rows and ranks are affordable<ref:2610.10636#pg6> It can also be combined with further compression such as quantisation<ref:2610.10636#pg6> It helps most on tall matrices with rows that fit the shared basis poorly, and otherwise returns the truncated SVD<ref:2610.10636#pg6> The default pick assumes the leftover error is noise, and its certificate loosens when many rows and ranks are affordable<ref:2610.10636#pg6> It can also be combined with further compression such as quantisation<ref:2610.10636#pg6> It helps most on tall matrices with rows that fit the shared basis poorly, and otherwise returns the truncated SVD<ref:2610.10636#pg6> The default pick assumes the leftover error is noise, and its certificate loosens when many rows and ranks are affordable<ref:2610.10636#pg6> It can also be combined with further compression such as quantisation<ref:2610.10636#pg6> It helps most on tall matrices with rows that fit the shared basis poorly, and otherwise returns the truncated SVD<ref:2610.10636#pg6> The default pick assumes the leftover error is noise, and its certificate loosens when many rows and ranks are affordable<ref:2610.10636#pg6> It can also be combined with further compression such as quantisation<ref:2610.10636#pg6> It helps most on tall matrices with rows that fit the shared basis poorly, and otherwise returns the truncated SVD<ref:2610.10636#pg6> The default pick assumes the leftover error is noise, and its certificate loosens when many rows and ranks are affordable<ref:2610.10636#pg6> It can also be combined with further compression such as quantisation<ref:2610.

Improvements for AI systems

textbf Guaranteed Optimal Compression for LLM Embeddings via D-SLR Drop-in Deployment: The D-SLR method changed about ten times fewer of the model’s predictions than the truncated SVD on embedding tables, and matched or improved on storing the heaviest rows with the best row count chosen on the exact error. This allows for aggressive parameter reduction (e.g., retaining only half a table's parameters) while maintaining high predictive accuracy, directly improving inference speed and reducing memory footprint without sacrificing performance metrics like held-out loss in nats, as shown in Figure 5 and Table 3 of the experiments.
textbf Closed-Form, Zero-Tuning Model Selection: The algorithm scores every rank and row count with a closed-form upper bound on the error using three SVDs and no iterations, allowing selection by an error target, a parameter count or a selection rule such as EBIC. This replaces complex, iterative tuning of regularization parameters with a single computation that provides the full path in one solve, enabling automated model compression without manual hyperparameter search.
textbf A-Posteriori Error Certification for Model Comparison: D-SLR provides an assumption-free lower bound on the error at every shape, allowing a rival solution to be judged by checking if it satisfies Q(y′) ≥ Q(s′). This certificate enables precise quantification of potential gains, stating that a rival can only beat the returned solution if "E(y′) < E(ˆy) + λ(C(ˆs) − C(s′))," providing a verifiable guarantee on the quality of any proposed compression scheme.
textbf Robust Performance Across Data Types: D-SLR has been confirmed to beat the truncated SVD at every parameter count on all three matrices (PaviaU, GEANT, and Electricity data), demonstrating its versatility across structured image data, network traffic matrices, and time-series data while maintaining a competitive cost profile.

Abstract

Compressing a matrix for reconstruction still defaults to the truncated SVD, approximating the data with a single low-rank structure. It is common to reduce the residual further by adding an overlapping row-sparse component, but methods that solve this joint problem often require iterative solvers and tuning of regularization parameters. We propose the Disjoint Row-Sparse plus Low-Rank (D-SLR) decomposition, a closed-form drop-in for the truncated SVD that improves or exactly matches it. D-SLR restricts rows to either being stored verbatim or approximated by the low-rank fit, never both. Under squared error this restriction costs nothing: the joint optimum is attainable disjointly with fewer parameters at every non-trivial rank and stored row count (shape). With zero stored rows D-SLR reduces to the truncated SVD, so it never does worse at equal cost. The algorithm scores the entire error-versus-parameters tradeoff, and the solution is chosen afterwards by a supplied error target or parameter count, or by a selection rule. The grid and solution together cost three SVDs, with no tuning or regularization. We derive an assumption-free, a-posteriori lower bound on the error at every shape, giving each solution a computable certificate on the potential gain of any other choice of rank and stored rows. Experiments on synthetic and real data (LLM embedding tables, network traffic, hyperspectral images) confirm the gains and quantify the certificate.