Learning to Rank

The RISC-V Vector Extension (RVV) provides SIMD-style vector execution for accelerating deep learning (DL) inference on edge processors. However, existing low-rank compression methods mainly select ranks according to accuracy preservation and theoretical floating-point operation (FLOP) reduction, without considering whether the selected ranks are profitable under a target RVV effective vector length. As a result, a low-rank factorization that reduces FLOPs may still introduce tail-handling overhead or even increase the RVV instruction count under specific vector configurations. To address this problem, we propose VALOR, a low-rank restructuring framework for efficient DL deployment on RISC-V processors. Given a pretrained model and a target RVV configuration, VALOR identifies decomposable layers in the pretrained model, filters rank candidates using an RVV instruction-profitable condition, and searches for a global restructuring strategy with minimal accuracy loss. Furthermore, to avoid repeated end-to-end evaluation, VALOR uses a Hessian-aware accuracy predictor that combines layer-wise factorization error with inter-layer coupling. Experiments on Spike and a real RISC-V vector processor, SpacemiT X60, show that VALOR achieves a better accuracy-efficiency trade-off than baseline methods.

Adapting Large Language Models (LLMs) to downstream tasks is increasingly bottlenecked by the staggering memory overhead of first-order (FO) backpropagation. Zeroth-order (ZO) optimization has emerged as a compelling, memory-efficient alternative; however, it suffers from the curse of dimensionality, which introduces prohibitively high variance in gradient estimation. While LLM gradients empirically reside in low-rank subspaces, existing state-of-the-art ZO methods typically rely on randomly generated low-rank subspaces, which often fail to align with the underlying dominant gradient manifold, fundamentally limiting optimization efficiency. To bridge this gap, we propose AHZO, an efficient low-rank ZO fine-tuning algorithm that dynamically identifies dominant subspaces via the Average Historical Gradient (AHG). Motivated by the directional coherence of optimization trajectories, AHZO leverages the average historical ZO gradient over a period as a principled proxy for the true gradient. By applying singular value decomposition to AHG matrices, AHZO distills principal spectral signatures to construct low-rank bases that tightly align with the true dominant subspaces. Theoretically, we prove that AHZO significantly mitigates estimation variance, yielding superior convergence guarantees compared to random-subspace ZO methods. Extensive evaluations across diverse LLM architectures and benchmarks demonstrate that AHZO consistently outperforms existing ZO baselines, achieving performance highly competitive with memory-intensive FO fine-tuning.

Training neural networks directly in a low-rank parameterization is an appealing route to reducing memory, compute, and storage simultaneously during both training and inference. Dynamic low-rank training (DLRT), which confines weights to a rank-$r$ manifold via the Galerkin projection of the gradient flow, is particularly attractive because it identifies efficient subnetworks on the fly without specialized initialization or post-factorization. However, DLRT fails to find trainable networks under high compression. In this paper, we derive the gradient flow of the best rank-$r$ approximation and point out that the offset of DLRT comes from a curvature-coupling term which is large and thus non-negligible under aggressive compression. Guided by this analysis, we propose a stable dynamic low-rank training method, named SDLRT, which maintains a lightweight compensation buffer that reinjects the top neglected singular directions. Additionally, we introduce a negative feedback on the truncation tolerance to stabilize each layer's rank. Experimentally, SDLRT reliably finds trainable subnetworks where DLRT collapses and as a PEFT adapter on DeBERTa-v3, it achieves the best average score on SuperGLUE at only $2.8\\%$ parameter overhead over LoRA.

Aligning Large Language Models (LLMs) with human preferences typically relies on external supervision, which faces critical limitations: human annotations are scarce and subjective, reward models are vulnerable to reward hacking, and self-evaluation methods suffer from prompt sensitivity and biases. In this work, we propose **stable rank**, an intrinsic, annotation-free quality signal derived from model representations. Stable rank measures the effective dimensionality of hidden states by computing the ratio of total variance to dominant-direction variance, capturing quality through how information distributes across representation dimensions. Empirically, stable rank achieves 84.04\% accuracy on RewardBench and improves task accuracy by an average of 14.1\% relative to greedy decoding via Best-of-N sampling. Leveraging this insight, we introduce **Stable Rank Group Relative Policy Optimization (SR-GRPO)**, which uses stable rank as a reward signal for reinforcement learning. Without external supervision, SR-GRPO improves Qwen2.5-1.5B-Instruct by 10\% on STEM and 19\% on mathematical reasoning, outperforming both learned reward models and self-evaluation baselines. Our findings demonstrate that quality signals can be extracted from internal model geometry, offering a path toward scalable alignment without external supervision.

The growing context lengths of large language models (LLMs) have made key-value (KV) cache memory a critical bottleneck during inference. Vector quantization (VQ) with commutative codebooks provides an effective Rotary Position Embedding (RoPE)-aware solution by enabling reconstruction-free attention over quantized key indices. However, existing commutative VQ methods still parameterize scalar codebook coefficients across RoPE subspaces and codewords largely independently, leaving parameter-side redundancy underexploited. We propose $\textbf{RankVQ}$, a low-rank parameterized commutative vector quantization method for KV cache compression. Instead of factorizing the deployed codebooks, RankVQ factorizes the underlying scalar coefficient matrices used to construct key and value codebooks. This design preserves the pseudo-symmetric structure required for RoPE commutativity on the key side, while improving the quality and compactness of offline codebook construction. The low-rank factors are discarded after codebook construction, so RankVQ keeps the same online inference form as standard commutative VQ and introduces no additional decoding overhead. Experiments on LongBench, InfiniteBench, and GSM8K show that RankVQ achieves a stronger accuracy--compression trade-off than directly comparable KV quantization baselines, with particularly clear gains in the 1-bit regime.

Single-step inorganic retrosynthesis is evaluated by whether a complete precursor set is recovered, yet many data-driven systems first rank individual precursors and then rely on threshold or count heuristics to assemble sets. This mismatch is especially severe when the true set size is unknown: individually plausible precursors can combine into incomplete, over-complete, or otherwise wrong near-miss sets. We reformalize the task as variable-size set-level ranking and introduce RECIPE, a target-conditioned framework that separates precursor recall from final set-level scoring. The Precursor Candidate Generator learns formula-level compatibility to build a high-recall precursor pool, while the Complete-Set Reranker compares variable-size candidate precursor sets directly. On the Retrieval-Retro year-split benchmark, the generator improves Combo@20 from 69.00 to 72.75. Compared with Retrieval-Retro, the reranker improves Combo@1 by 11.30 points to 71.70, Combo@20 by 20.82 points to 89.82, and Combo MRR by 14.14 points to 77.43. A preliminary 20-case out-of-distribution evaluation shows the same direction of improvement. These results suggest that set-level prioritization remains important after strong precursor recall, and that optimizing ranked complete precursor sets can reduce the inspection burden in inorganic synthesis planning.

We study representation learning in low-rank Constrained Markov Decision Processes (CMDPs), where both value and reward functions are approximated by a set of unknown representation vectors, and the transition dynamics admit a low-rank factorization. The objective is to maximize expected cumulative rewards while complying with constraints on the expected cumulative utility. To achieve this, we propose \texttt{MFRLC} (Model-Free Representation Learning for Low-Rank CMDP), a model-free algorithm that uses a primal-dual scheme to effectively balance reward regret and constraint violations. To the best of our knowledge, \texttt{MFRLC} is the first representation-learning method for low-rank CMDPs that combines the Least-Squares Value Iteration with Upper Confidence Bound (LSVI-UCB) and primal-dual techniques. \texttt{MFRLC} further enhances value function estimation with bonuses for function approximation and uniquely learns the mapping from representations to value functions directly through representation learning. We prove that \texttt{MFRLC} achieves both regret and cumulative constraint violation of order $\widetilde O(H^3 d^2 K^{3/4}|\mathcal{A}|^{3/2}/\gamma)$, making it provably sample-efficient and highly adaptable to complex environments due to the reliance on function approximation.

Backpropagation (BP) is widely viewed as biologically implausible, in part because it requires feedback weights to be the transpose of forward weights for error propagation. Interestingly, when training a network with fixed random feedback weights to circumvent this issue, learning aligns the forward weights with the feedback weights, leading the backpropagated error signal to become an approximation of the standard gradient used by BP. This process, called Feedback Alignment (FA), occurs in MLPs and very shallow CNNs but does not scale well to deeper architectures. In this work, we first investigated differences between BP and FA models, trained on CIFAR10, specifically focusing on the effective rank of the signal. We found that the FA error has a considerably lower rank and hence is constrained to a lower-dimensional subspace compared to BP, limiting exploration of the parameter space. Motivated by this observation, we evaluated two mechanisms for increasing the effective dimensionality of FA: Muon, an optimiser that orthogonalises weight updates; and hidden activity normalisation, which promotes activation orthogonality. Across larger architectures and benchmarks, we find that these methods consistently improve over FA baselines, for example, on CIFAR100 with a Resnet-18, accuracy increases by 9 percentage points. Our results identify low-dimensional gradient dynamics as a key obstacle to scaling FA and suggest that inducing higher-dimensional update geometry is a promising route toward scaling alternatives to backpropagation.

Unit-modulus measurement vectors, due to their phase-only variations, exhibit favorable hardware compatibility and storage efficiency. In this paper, we consider matrix recovery using symmetric unit-modulus rank-one measurements. We identify a fundamental limitation of symmetric unit-modulus measurements---their inability to capture individual diagonal entries, and show that matrix recovery can be achieved using such measurements when the diagonal entries are known or directly measurable. We construct a stacked operator and establish that it satisfies a mixed-norm restricted isometry property ($\mathrm{mRIP}$) for unit-modulus measurements. Leveraging the $\mathrm{mRIP}$ condition, we derive exact and stable recovery results for matrix recovery with unit-modulus measurements in both noiseless and noisy settings. Our work is the first to provide the theoretical foundation for matrix recovery under symmetric rank-one unit-modulus measurements. Numerical experiments further corroborate these theoretical findings.

The quality of open-weight language models has dramatically improved in recent years. Sharing weights greatly facilitates model adoption by enabling their use across diverse hardware and software platforms. They also allow for more open research and testing, to the extent that users can use them as checkpoints, fine-tune them according to their needs, and potentially redistribute them. In some cases, however, concerns on modifying these weights towards unauthorized uses may outweigh the pros of giving users such a freedom. Defending against such adaptation is non-trivial: since an adaptive attacker can observe all weights and architectures by definition, they can reverse simple structural defenses, and use optimization to defeat the simplest locking mechanisms. In this work, we exploit the inference–training asymmetry of automatic differentiation as a novel defense axis. We propose DLR-Lock, a method where the purveyor of the model purposely replaces each pretrained MLP in their model with a deep low-rank residual network (DLR-Net) of comparable parameter count, forcing activation memory that grows linearly with depth during backpropagation. DLR-Nets are efficiently trained via module-wise distillation. We show that, beyond this memory overhead, DLR-Lock results in architectural mismatches that complicate the optimization landscape of standard fine-tuning, and a backward pass that incurs disproportionately more overhead than the forward pass. Our defense succeeds in withstanding adaptive attackers with full knowledge of the defense strategy while preserving the original model's capabilities. Experiments on LLM validate these claims.

We study the sequential decision-making problem where at each time the learner selects an assortment of bounded size and receives a ranking of the selected items. Ranking feedback arises naturally when human evaluators or automated LLM judges compare multiple options simultaneously. It is often easier and more reliable than assigning absolute scores, which are typically poorly calibrated across evaluators, while yielding substantially more information than a single pairwise comparison. We focus on rankings that are constructed according to the random utility model with linear utilities, which includes the popular Plackett-Luce model as a special case. We show that improved regret minimisation is possible under full rank breaking with a refined analysis based on 1-factorisations and Baranyai's theorem that fully exploits independence in the pairwise comparisons. This gives a tighter confidence interval around the true utility parameter which is crucial in our analysis for regret that improves with maximum assortment size. Furthermore, it matches the regret lower bound for our model in case the learner is permitted to play multi-sets of the maximum bounded size. Our work gives a principled approach to learning from ranking feedback that shows consistent improvement with the maximum assortment size.

Learning in neural systems arises from synaptic changes that reshape the representations underlying behavior. While low-rank recurrent neural networks (RNNs) have emerged as a powerful framework for linking connectivity to function, a theoretical understanding of their learning process remains elusive. Here, we extend the low-rank framework from activity to learning by deriving gradient-descent dynamics directly in a reduced overlap space. We formulate a closed-form, low-dimensional system of ODEs that governs learning in this space, exact for linear RNNs and asymptotically exact for nonlinear RNNs in the large-$N$ Gaussian limit. Central to our analysis is a distinction between two classes of overlaps: _loss-visible_ overlaps, which fully determine network activity, output, and loss, and _loss-invisible_ overlaps, which do not affect function but are required to describe learning. We illustrate the consequences of this decomposition through two phenomena. First, we show that learning can serve as a perturbation that exposes differences in connectivity between functionally equivalent networks. Second, we show that loss-invisible overlaps can act as memory variables that encode training history, and characterize the conditions under which this occurs. Finally, we present several testable predictions for biological learning experiments derived from our theory.

Softmax attention in Transformers suffers from quadratic complexity in sequence length, making it impractical for long-context applications. Linear attention alleviates this issue by replacing the exponential kernel with alternative functions that enable linear-time computation. Among existing linear attention approaches, recent studies have shown that polynomial kernels are particularly effective, as they exhibit sparse and spiky behavior similar to softmax attention, which emphasizes large dot products while suppressing irrelevant interactions. However, the exact computation of high-degree polynomial kernels is infeasible for high-dimensional representations. As a result, prior work relies on approximate polynomial kernels. This introduces a non-negligible approximation error. In this paper, we show that, from the perspective of polynomial kernel approximation, existing linear attention methods are still suboptimal. We propose Learnable Low-Rank Polynomial Sketch (LLoPS), a principled and flexible framework for approximating polynomial kernels with linear attention. Our method learns a low-rank polynomial sketch that provably achieves a strictly smaller approximation error than existing approaches. Experiment results show that LLoPS achieves the highest performance across extensive benchmarks, comparing with various linear attention baselines.

State-of-the-art Extreme Multi-Label Text Classification (XMC) models rely on multi-label attention to focus on key tokens in input text, but learning high-quality attention weights is challenging. We introduce PLANT (Pretrained and Leveraged Attention), a plug-and-play strategy for initializing attention. PLANT works by \emph{planting} label-specific attention using a pretrained Learning-to-Rank model guided by mutual information gain. This architecture-agnostic approach integrates seamlessly with large language model backbones such as Mistral, LLaMA, DeepSeek, and Phi-3. PLANT outperforms state-of-the-art methods across tasks such as ICD coding, legal topic classification, and content recommendation. Gains are especially pronounced in few-shot settings, with substantial improvements on rare labels. Ablation studies confirm that attention initialization is a key driver of these gains. Code and trained models are available at \url{https://github.com/research-anon-487/xcube/tree/plant}.

Post-training quantization (PTQ) to FP4 has emerged as a key technique for reducing large language model inference cost, especially on Blackwell GPUs which offer native FP4 tensor core support. Direct PTQ to FP4, however, still leaves a noticeable accuracy gap. Recent work on SVDQuant mitigates the errors introduced through FP4 quantization for diffusion models by adding high-precision low-rank corrections obtained via SVD. However, this high-precision low-rank correction requires many bits per rank, and moreover competes on tensor core utilization with the main FP4 GEMM. We introduce BiLoCo, a correction whose left and right factors are stored as binary ${\{\pm 1\}}$ sign vectors rather than as BF16 numbers. The signed format uses $16{\times}$ less storage per rank-one term, and since the correction now consists of additions/subtractions, this can be run on CUDA cores concurrently with the FP4 tensor-core GEMM. On 4--32B LLMs, BiLoCo matches or improves memory-matched SVDQuant for PTQ. We implement BiLoCo on B200 with a correctness-preserving CUDA schedule that runs end-to-end decode at $1.13$--$1.15\times$ FP4-only, with the residual gap coming from the final output combine rather than from one-bit arithmetic.

Parameter-efficient fine-tuning of quantized large language models is dominated by QLoRA, which combines 4-bit weight quantization with FP16 low-rank adapters. We show the FP16 adapter precision is unnecessary: a single binary bit per adapter parameter, combined with a learned per-layer scale $s$ and a Scaled Straight-Through Estimator, matches or exceeds QLoRA across a broad range of benchmarks while reducing adapter storage by $14$--$16\times$. We introduce two variants--NormScale, which clips STE gradients to the unit ball, and LearnThresh, which makes the binarization threshold learnable---unified under a framework we call BQ-LoRA. Our main theoretical contribution is a tight characterization of the BQ-LoRA hypothesis class: it has bounded entrywise norm, a high-probability spectral norm bound that is $\sqrt{r}$ times sharper than the worst case, a generically full-rank update, and consequently a Rademacher complexity that yields an explicit $\tilde{O}(s_{\max}\sqrt{d_{\text{in}} d_{\text{out}}/n})$ generalization bound. None of these properties hold a priori for FP16 LoRA. On Qwen2.5 (1.5B/7B/72B) and LLaMA-2 (7B/70B), BQ-LoRA matches or beats QLoRA and LoftQ at $\sim 1/16$ the adapter footprint; the predicted regularization signature appears empirically in dataset ablations and training-loss curves. We extend Punica's SGMV kernel with a binary inner-product primitive (BQ-SGMV) and prove a cache phase-transition result: shrinking adapters $16\times$ enlarges the HBM cache from $2{,}500$ to $39{,}000$ adapters, eliminating misses under skewed workloads.

Low-rank factorization is a standard way to scale optimization in machine learning by replacing large matrix variables with compact factors. For positive semidefinite (PSD) variables, the symmetric Burer--Monteiro factorization (sBMF) writes $Z=XX^\top$ with a single low-rank factor $X$. A recent asymmetric alternative (aBMF) writes $Z=XY^\top$ and adds a quadratic penalty $(\gamma/2)\|X-Y\|_F^2$ to encourage symmetry. This split is attractive because it yields a biconvex objective with alternating convex subproblems, but its practical value depends strongly on how the penalty parameter $\gamma$ is chosen. We study a unified regularized aBMF framework and derive an explicit lower bound on $\gamma$ that guarantees exactness: under mild assumptions, any $\gamma$ above this threshold makes aBMF and sBMF share the same critical points. This gives a principled way to use the asymmetric formulation without altering the critical-point structure of the symmetric problem. In particular, it answers the open question of whether an exact penalty exists for asymmetric relaxation.

Low-rank plus sparse decompositions, such as robust PCA, typically model the low-rank component as dense and the sparse component as corruption. This viewpoint is poorly suited to graph adjacency matrices, where sparsity is structural and dense low-rank factors destroy the combinatorial and computational properties of the graph. We propose signed biclique decomposition (SBD), a sparse low-rank decomposition for integer-weighted graphs that represents the structured component as a sum of sparse signed binary rank-one factors, together with a sparse signed residual. The resulting representation remains discrete, interpretable, sparse, and compatible with standard linear algebra. We formulate fixed-rank and greedy SBD objectives and derive a gain-based greedy algorithm with theoretical guarantees. Empirically, SBD recovers block structure in noisy graphs, while its factorized representation enables efficient sparse matrix-matrix multiplication (SpMM). On real graphs with strong shared-neighborhood structure, SBD achieves compression factors above $10$ and SpMM speedups up to $6\times$ over cuSPARSE, suggesting a practical alternative to dense low-rank decompositions for graphs.