Train What You Deploy: Closing the MLP Reachability Gap in Low-Rank Clone Distillation

A compressed student has two shapes that need not agree: the weight it deploys at inference and the weight family its training can reach. We show that a state-of-the-art weight-inheritance distiller, Low-Rank Clone (LRC), deploys a full-width student MLP but ties training to a teacher-induced slice, leaving 62.5-81.4% of each deployed matrix's independent linear degrees of freedom unreachable-paid for at inference, never trainable. Our principle is one line: train what you deploy. From the identical LRC warm start, we make the training object the entire deployed matrix, with no change in deployed shape, deployed parameter count, or inference FLOPs, via two mergeable realizations (Dense-LRC and CORE-LRC) that both collapse to one deployed weight. This recovers stranded capacity: taking the stronger realization per teacher, +2.36/+2.71/+10.45 Avg9 over matched-budget plain-LRC baselines across three teachers (Llama3.2-3B, Llama3.1-8B, Qwen2.5-3B), with the largest gain on the widest teacher (Qwen), where it reaches the original recipe's approx. 20B-token accuracy at 10B tokens (2x token efficiency); there the strictly same-lineage arm still recovers +6.39, the fully controlled figure. Controls strongly support attributing the gain to the enlarged reachable set, rather than to added parameters or the recipe. From approx. 10B distillation tokens plus a short SFT, a half-parameter 1.5B student matches its approx. 9T-token teacher's 9-task macro-average, within evaluation noise and with a residual MMLU deficit, and a 2.7B student beats Meta's own official compression of Llama3.1-8B at ~900x fewer compression tokens (a token count under unmatched recipes, not a compute claim). All results are from single-seed runs on the LRC backbone.

Subzero matrix completion for sparse data analysis: large-scale learning of latent low-rank structure

We investigate when a sparse nonnegative matrix can be recovered from a real-valued matrix of much lower rank by zeroing out its negative elements. The potential for such decompositions suggests a mathematical connection between sparsity and rank; we analyze a number of sparse matrices with this latent low-rank structure and use them to illustrate the geometric origins of this connection. Previous algorithms have discovered these decompositions via an alternating minimization over the factors of a low-rank matrix, but to do so, they have also needed to compute and store another matrix, neither sparse nor low-rank, that is the size of their product. We develop a stochastic, alternating least-squares algorithm that operates on smaller blocks of this dense matrix and scales as a result to much larger problems. We also show how to further accelerate this algorithm with sparse optimizations and customized CUDA kernels. As one example, we use the algorithm to analyze the sparse matrix of synaptic weights for the recently published $\textit{Drosphilia}$ connectome. The nonzero elements of this matrix, with 139,255 rows and columns, record the number of synapses between cells in the nervous system of a female fruit fly. Despite a slowly decaying spectrum of singular values, this matrix exhibits a latent low-rank structure that is predictive of cell categories across multiple levels of specificity.

MaLViL: Multi-axis Low-rank Vision-LSTM for Medical Image Segmentation

Vision-LSTM (ViL) enables efficient global modeling, but its cost still scales with the number of spatial tokens, so existing segmenters confine ViL to a coarse bottleneck and lose fine anatomical detail. Rasterizing 2D features into a 1D sequence further breaks adjacency across the orthogonal scan axis. We propose MaLViL, a Multi-axis Low-rank Vision-LSTM network that extends ViL across decoder resolutions. Bidirectional low-rank ViL (Bi-LRViL) reasons on a compact orthonormal subspace and preserves detail through an orthogonal residual; scale-aware SaLViL restores cross-axis neighbors before serialization; and a Cross-Directional Mixer (CDM) fuses orthogonal horizontal and vertical traversal paths. Statistics-Guided Skip Modulation (SGSM) further retains boundary cues in encoder skips. On skin-lesion, ultrasound, and multi-organ CT benchmarks, MaLViL achieves competitive or state-of-the-art segmentation accuracy, while reducing ViL operator memory by up to $83\times$ at fine decoder resolutions. Code is available at: https://github.com/xmindflow/malvil.

K-space Gaussian Representation for Parallel MRI

Accelerated magnetic resonance imaging (MRI) aims to recover the k-space signal from acquired measurements, where accurate estimation of missing samples is essential for high-fidelity reconstruction. Existing k-space reconstruction methods estimate missing samples through interpolation operators or structure priors defined on discrete sampling grids. Although these formulations effectively exploit local interpolation relationships and global k-space redundancy, they reconstruct only discrete frequency coefficients and therefore do not explicitly model the underlying continuous signal. To overcome this limitation, we propose K-space Gaussian Representation (KGR), the first explicit continuous representation formulated directly in the native k-space domain. Rather than estimating unknown samples on discrete grids, KGR parameterizes the continuous signal using Gabor-Gaussian primitives with shared spatial geometry, yielding a compact representation that naturally preserves inter-coil correlations. Because unconstrained continuous fitting does not necessarily satisfy the intrinsic structural properties of multi-coil signal, the estimated representation is projected onto a low-rank manifold to enforce the algebraic constraints arising from smoothly varying phase and coil redundancy. A frequency-adaptive fitting strategy accommodates the heterogeneous characteristics of different k-space regions. Comprehensive validation across multiple datasets and sampling schemes shows consistent improvements over representative reconstruction baselines in both quantitative metrics and visual quality. These results suggest that explicit continuous parameterization of native k-space provides a principled framework for integrating continuous signal modeling with structured low-rank reconstruction.

Graph-Regularized Low-Rank Matrix Completion by Variable Projection

We address the low-rank matrix completion problem by incorporating graph regularization into the existing Riemannian Trust-Region Matrix Completion (RTRMC) framework. The latter uses the geometry of the low-rank constraint to remodel the problem as an unconstrained optimization problem on a single Grassmann manifold. Our approach, named Graph-Regularized RTRMC (GR-RTRMC), exploits the inherent relationships between rows and columns of the matrix. By using these relationships, we aim to improve the accuracy and robustness of matrix completion, particularly in scenarios where the underlying data exhibits strong correlations between rows or columns.

Low-Rank Tensor Completion Based on Fractional Regularization with Ky Fan p-k Norm

This paper addresses low-rank tensor completion (LRTC) by proposing a novel nonconvex surrogate, namely the ratio of the tensor nuclear norm to the tensor Ky Fan p-k norm (TNPK), to accurately approximate the tensor tubal rank. The TNPK possesses appealing properties, including scale invariance, parameter flexibility, and the existence of closed-form solutions under specific choices of p and k. With specific parameter settings of p and k, it reduces to the ratio of the tensor nuclear norm to the tensor Ky Fan k norm (TNK) or the ratio of the tensor nuclear norm to the tensor Frobenius norm (TNF). We construct a LRTC model and, under the tensor null space property (NSP), prove that low-rank tensors are local minimizers of the proposed model. Moreover, we derive the proximal operator of the Ky Fan p-k inverse-norm and further develop an efficient alternating direction method of multipliers (ADMM) algorithm with guaranteed subsequential convergence under mild conditions. Extensive experiments on synthetic and real-world datasets validate the superior performance of our method against state-of-the-art competitors.

Low-rank Distributional Matrix Completion

We study a distributional generalization of the matrix completion problem in which each entry of the target matrix is a probability distribution rather than a scalar. In this setting, only a subset of matrix entries is observed, and even for observed entries, the underlying distributions are not directly accessible; instead, we observe finitely many samples drawn from them. To represent distributional entries, we employ kernel mean embeddings and introduce a notion of Tucker rank for distribution-valued matrices to capture their low-rank structure. The infinite-dimensional nature of kernel embeddings poses significant methodological challenges. To address this, we introduce functional unfolding operators that link the proposed distributional low-rank structure to the classical Tucker rank for finite-dimensional tensors. Based on this framework, we propose a novel estimator for distributional matrix completion. We establish non-asymptotic error bounds that characterize the statistical performance of the estimator. Extensive experiments on synthetic data and a real-world application demonstrate the effectiveness of the proposed method.