Daily Papers

The field of texture synthesis has witnessed important progresses over the last years, most notably through the use of Convolutional Neural Networks. However, neural synthesis methods still struggle to reproduce large scale structures, especially with high resolution textures. To address this issue, we first introduce a simple multi-resolution framework that efficiently accounts for long-range dependency. Then, we show that additional statistical constraints further improve the reproduction of textures with strong regularity. This can be achieved by constraining both the Gram matrices of a neural network and the power spectrum of the image. Alternatively one may constrain only the autocorrelation of the features of the network and drop the Gram matrices constraints. In an experimental part, the proposed methods are then extensively tested and compared to alternative approaches, both in an unsupervised way and through a user study. Experiments show the interest of the multi-scale scheme for high resolution textures and the interest of combining it with additional constraints for regular textures.

Representational similarity methods compare the geometries of neural representations, but they do not measure how consistently the geometry of a single representation is recovered from subsets of its feature coordinates. We call this property geometric stability and introduce Shesha, which estimates it by correlating representational dissimilarity matrices from complementary random feature subsets. Shesha is not invariant to orthogonal rotations: representations with identical Gram matrices, and therefore identical linear CKA, can have different geometric stability. Controlled transformations further separate the quantities. Across 2{,}463 encoder configurations spanning seven domains, similarity and stability are positively associated across non-PCA transformations (ρ=+0.75) but negatively associated under PCA-coordinate compression (ρ=-0.47). We further evaluate 170 pretrained vision models across six datasets. DINOv2 combines strong transfer performance with bottom-quartile stability on five of six datasets, showing that transferability and feature-split stability need not coincide. Across random feature subsets, the marginal relationship between Shesha and linear-probe variability is dataset-dependent; after controlling for task alignment with LogME, higher Shesha is associated with lower variability on five of six datasets. These results identify geometric stability as a basis-dependent property that complements representational similarity and task alignment.

Structured deep model compression methods are hardware-friendly and substantially reduce memory and inference costs. However, under aggressive compression, the resulting accuracy degradation often necessitates post-compression finetuning, which can be impractical due to missing labeled data or high training cost. We propose post-hoc blockwise compensation, called GRAIL, a simple zero-finetuning step applied after model compression that restores each block's input-output behavior using a small calibration set. The method summarizes hidden activations via a Gram matrix and applies ridge regression to linearly reconstruct the original hidden representation from the reduced one. The resulting reconstruction map is absorbed into the downstream projection weights, while the upstream layer is compressed. The approach is selector-agnostic (Magnitude, Wanda, Gram-based selection, or folding), data-aware (requiring only a few forward passes without gradients or labels), and recovers classic pruning or folding when the Gram matrix is near identity, indicating weak inter-channel correlations. Across ResNets, ViTs, and decoder-only LLMs, GRAIL consistently improves accuracy or perplexity over data-free and data-aware pruning or folding baselines in practical compression regimes, with manageable overhead and no backpropagation. The code is available at https://github.com/TWWinde/GRAIL_Compensation.

We examine how closely pretrained LeWorldModel (LeWM) reproduces the effects of small action changes. Using matched simulator and model responses, we measure action sensitivity with Gram matrices at three perturbation scales. Across nearly a thousand states from Reacher and OGBench Cube, the most sensitive action directions differ by median angles of about 36^circ and 42^circ, respectively, at the smallest scale. Increasing the perturbation tenfold reduces median normalized Gram discrepancy by roughly 70\% on Reacher and 47\% on Cube. Geometry measured in pixels and embeddings varies more across scales than the chosen physical-state geometry. On held-out directions, environment Grams achieve median embedding response errors of 17\% and 28\%, compared with 26\% and 38\% for an isotropic reference with the same trace. LeWM's errors are higher at this scale. These findings support scale-specific evaluation of action sensitivity; its effect on control remains an open question.

Let λ_{mathbb K}(m) denote the maximal absolute projection constant among m-dimensional Banach spaces over mathbb K=mathbb R or mathbb C. Its exact value is known only in a few cases, and determining it remains a challenging problem. In this note, we investigate several structured vector configurations that naturally arise in this context. We first recall the connection between maximal projection constants and tight frames, then consider biangular tight frames, and show that their relative and quasimaximal projection constants coincide. A particularly interesting example is provided by the midpoints of the edges of a regular simplex, which yield natural lower bounds for λ_{mathbb R}(m). Numerical evidence suggests that these bounds may be sharp in dimensions 6 and 8. We also discuss weighted spherical (2,2)-designs with a small number of vectors and their connection with maximal projection constants. Examples in dimensions 4 and 5 indicate that the sign patterns of their Gram matrices may play an important role. These observations lead us to formulate several conjectures concerning maximal absolute projection constants and the vector configurations associated with them.

Low-discrepancy (LD) sequences have been extensively used as efficient experimental designs across many scientific disciplines. QMCPy (https://qmcsoftware.github.io/QMCSoftware/) is an accessible Python library which provides a unified implementation of randomized LD sequences, automatic variable transformations, adaptive Quasi-Monte Carlo error estimation algorithms, and fast kernel methods. This article focuses on recent updates to QMCPy which broaden support for randomized LD sequences and add new tools to enable fast kernel methods using LD sequences. Specifically, we give a unified description of the supported LD lattices, digital nets, and Halton point sets, along with randomization options including random permutations / shifts, linear matrix scrambling (LMS), and nested uniform scrambling (NUS). We also support higher-order digital nets, higher-order scrambling with LMS or NUS, and Halton scrambling with LMS or NUS. For fast kernel methods, we provide shift-invariant (SI) and digitally-shift-invariant (DSI) kernels, including a new set of higher-order smoothness DSI kernels. When SI and DSI kernels are respectively paired with n LD lattice and digital net points, the resulting Gram matrices permit multiplication and inversion at only O(n log n) cost. These fast operations utilize QMCPy's implementation of the fast Fourier transform in bit-reversed order (FFTBR), inverse FFTBR (IFFTBR), and fast Walsh--Hadamard transform (FWHT).

Multimodal large language models (MLLMs) incur substantial inference cost due to the processing of hundreds of visual tokens per image. Although token pruning has proven effective for accelerating inference, determining when and where to prune remains largely heuristic. Existing approaches typically rely on static, empirically selected layers, which limit interpretability and transferability across models. In this work, we introduce a matrix-entropy perspective and identify an "Entropy Collapse Layer" (ECL), where the information content of visual representations exhibits a sharp and consistent drop, which provides a principled criterion for selecting the pruning stage. Building on this observation, we propose EntropyPrune, a novel matrix-entropy-guided token pruning framework that quantifies the information value of individual visual tokens and prunes redundant ones without relying on attention maps. Moreover, to enable efficient computation, we exploit the spectral equivalence of dual Gram matrices, reducing the complexity of entropy computation and yielding up to a 64x theoretical speedup. Extensive experiments on diverse multimodal benchmarks demonstrate that EntropyPrune consistently outperforms state-of-the-art pruning methods in both accuracy and efficiency. On LLaVA-1.5-7B, our method achieves a 68.2% reduction in FLOPs while preserving 96.0% of the original performance. Furthermore, EntropyPrune generalizes effectively to high-resolution and video-based models, highlighting the strong robustness and scalability in practical MLLM acceleration. The code will be publicly available at https://github.com/YahongWang1/EntropyPrune.

Communication-avoiding Krylov methods require solving small dense Gram systems at each outer iteration. We present a low-synchronization approach based on Forward Gauss--Seidel (FGS), which exploits the structure of Gram matrices arising from Chebyshev polynomial bases. We show that a single FGS sweep is mathematically equivalent to Modified Gram--Schmidt (MGS) orthogonalization in the A-norm and provide corresponding backward error bounds. For weak scaling on AMD MI-series GPUs, we demonstrate that 20--30 FGS iterations preserve scalability up to 64 GPUs with problem sizes exceeding 700 million unknowns. We further extend this approach to Algebraic MultiGrid (AMG) coarse-grid solves, removing the need to assemble or factor dense coarse operators

This paper makes a formal connection between two families of widely used matrix factorization algorithms in numerical linear algebra. One family consists of the Jacobi eigenvalue algorithm and its variants for computing the Hermitian eigendecomposition and singular value decomposition. The other consists of Gaussian elimination and the Gram-Schmidt procedure with various pivoting rules for computing the Cholesky decomposition and QR decomposition respectively. Both families are cast as special cases of a more general class of factorization algorithms. We provide a randomized pivoting rule that applies to this general class (which differs substantially from the usual pivoting rules for Gaussian elimination / Gram-Schmidt) which results in the same linear rate of convergence for each algorithm, irrespective of which factorization it computes. A second important consequence of this randomized pivoting rule is a provable, effective bound on the numerical stability of the Jacobi eigenvalue algorithm, which addresses a longstanding open problem of Demmel and Veseli\'c `92.

Adaptive rounding methods such as GPTQ, or equivalently Babai's nearest plane algorithm, round a real matrix to integers under a quadratic metric. They process the entries in a fixed order, one at a time, propagating each rounding error to the entries not yet processed through a triangular feedback matrix. We study the two-sided version of this task, in which fixed nonsingular basis matrices act on both the left and the right of the residual; the familiar one-sided case is the special case of an identity right basis. Vectorizing the matrix turns the two-sided objective into a quadratic metric whose Gram matrix is a Kronecker product, so the one-dimensional algorithm applies verbatim, but takes quartic time in the matrix dimension. We present GPTQ-2D, which produces the identical rounded matrix in cubic time. It rounds the entries anti-diagonal by anti-diagonal; entries on the same anti-diagonal are independent and are rounded in parallel.

Let Ω_n denote the class of n times n doubly stochastic matrices (each such matrix is entrywise nonnegative and every row and column sum is 1). We study the diagonals of matrices in Ω_n. The main question is: which A in Ω_n are such that the diagonals in A that avoid the zeros of A all have the same sum of their entries. We give a characterization of such matrices, and establish several classes of patterns of such matrices.

Human perception integrates multiple modalities, such as vision, hearing, and language, into a unified understanding of the surrounding reality. While recent multimodal models have achieved significant progress by aligning pairs of modalities via contrastive learning, their solutions are unsuitable when scaling to multiple modalities. These models typically align each modality to a designated anchor without ensuring the alignment of all modalities with each other, leading to suboptimal performance in tasks requiring a joint understanding of multiple modalities. In this paper, we structurally rethink the pairwise conventional approach to multimodal learning and we present the novel Gramian Representation Alignment Measure (GRAM), which overcomes the above-mentioned limitations. GRAM learns and then aligns n modalities directly in the higher-dimensional space in which modality embeddings lie by minimizing the Gramian volume of the k-dimensional parallelotope spanned by the modality vectors, ensuring the geometric alignment of all modalities simultaneously. GRAM can replace cosine similarity in any downstream method, holding for 2 to n modalities and providing more meaningful alignment with respect to previous similarity measures. The novel GRAM-based contrastive loss function enhances the alignment of multimodal models in the higher-dimensional embedding space, leading to new state-of-the-art performance in downstream tasks such as video-audio-text retrieval and audio-video classification. The project page, the code, and the pretrained models are available at https://ispamm.github.io/GRAM/.

We present a novel family of language model (LM) estimation techniques named Sparse Non-negative Matrix (SNM) estimation. A first set of experiments empirically evaluating it on the One Billion Word Benchmark shows that SNM n-gram LMs perform almost as well as the well-established Kneser-Ney (KN) models. When using skip-gram features the models are able to match the state-of-the-art recurrent neural network (RNN) LMs; combining the two modeling techniques yields the best known result on the benchmark. The computational advantages of SNM over both maximum entropy and RNN LM estimation are probably its main strength, promising an approach that has the same flexibility in combining arbitrary features effectively and yet should scale to very large amounts of data as gracefully as n-gram LMs do.

In a seminal paper in 1959, Marcus and Ree proved that every ntimes n bistochastic matrix A satisfies |A|_{F}^2leq max_{σin S_n}A_{i,σ(i)} where S_n is the symmetric group on {1, ldots, n}. Erdős asked to characterize the bistochastic matrices for which the equality holds in the Marcus--Ree inequality. We refer to such matrices as Erdős matrices. While this problem is trivial in dimension n=2, the case of dimension n=3 was only resolved recently in~bouthat2024question in 2023. We prove that for every n, there are only finitely many ntimes n Erdős matrices. We also give a characterization of Erdős matrices that yields an algorithm to generate all Erdős matrices in any given dimension. We also prove that Erdős matrices can have only rational entries. This answers a question of~bouthat2024question.

An Erdös matrix E is a bistochastic matrix whose sum of squares of entries (Frobenius norm squared) equals its maxtrace (maximum of all the σ-traces for permutations σ's). We characterize all Erdös E by the patterns of their zero entries; showing that each such skeleton has at most one E. We present an algorithm to find all ntimes n Erdös matrices, which finds them up to nleqslant 5 quickly and also size n=6. We further show some presently known RCDS matrices to be Erdös.

Analytic Continual Learning (ACL) offers a computationally efficient alternative to gradient-based approaches. Recent ACL methods are based on Recursive Least Squares (RLS) and have achieved the state-of-the-art results compared to other alternatives. However, they falter significantly in Class-Incremental Learning scenarios characterized by Long-Tailed distributions. While the ill-conditioning of the autocorrelation (Gram) matrix is a known limitation of RLS, we demonstrate that class imbalance exacerbates this issue into a distinct spectral pathology: "tail" classes suffer from severe spectral collapse, rendering their subspaces numerically indistinguishable from noise. Standard Ridge Regression (L_2) fails to address this effectively as it applies isotropic regularization - a uniform penalty that is insufficient to stabilize the tail without over-shrinking the head. To address this, we propose Geometry-Spectral Rectification (GSR), a theoretically grounded framework that treats long-tailed learning as a spectral regularization problem. Unlike standard isotropic regularization (Ridge) which uniformly penalizes all eigenvalues, GSR acts as an anisotropic spectral filter, selectively inflating the collapsed eigenvalues of tail classes. We construct a structured, data-dependent spectral perturbation matrix Δ that selectively inflates collapsed tail eigen-directions of the Gram matrix. Theoretical analysis proves that GSR guarantees an improved stable rank for the Gram matrix, ensuring numerical stability. Extensive experiments show that GSR establishes a new state-of-the-art for analytic CIL, offering a superior trade-off between computational efficiency and robust generalization in long-tailed settings.

Numerical Linear Algebra (NLA) has consistently played a vital role in advancing science by providing tools to solve fundamental problems encountered in scientific and engineering applications. Over the decades, it has continually evolved to meet the demands driven by successive waves of scientific discovery. For instance, during the 1950s and 1960s, substantial efforts were devoted to developing methods for solving eigenvalue problems that emerged from the rapidly growing field of aerodynamics. This led to the discovery of the LR and QR algorithms. Later the attention turned to the solution of sparse linear systems that were common in applications like computational aerodynamics. Today we are experiencing yet another wave of major scientific advancement and NLA is once more at the heart of its development. This Machine Learning (ML) wave is proving to be utterly disruptive in science and engineering. Many tools in ML particularly Large Language Models (LLMs) are grounded in matrix and tensor methods. As we are approaching Artificial General Intelligence (AGI), it is clear that matrix methods will be called to play an even more significant role. For the numerical linear practitioner the speed of the current change makes it particularly challenging to adapt. This is a survey article that centers on machine learning techniques, with a particular focus on large language models. It has two main objectives. The first is to clarify the core concepts behind Large Language Models in a manner accessible to specialists in numerical methods. The second is to examine the key Numerical Linear Algebra concepts employed by LLM techniques, while also highlighting several significant recent contributions of NLA to the field.

In this paper, we propose a new class of metric for table structure recognition (TSR) evaluation, called grid table similarity (GriTS). Unlike prior metrics, GriTS evaluates the correctness of a predicted table directly in its natural form as a matrix. To create a similarity measure between matrices, we generalize the two-dimensional largest common substructure (2D-LCS) problem, which is NP-hard, to the 2D most similar substructures (2D-MSS) problem and propose a polynomial-time heuristic for solving it. This algorithm produces both an upper and a lower bound on the true similarity between matrices. We show using evaluation on a large real-world dataset that in practice there is almost no difference between these bounds. We compare GriTS to other metrics and empirically validate that matrix similarity exhibits more desirable behavior than alternatives for TSR performance evaluation. Finally, GriTS unifies all three subtasks of cell topology recognition, cell location recognition, and cell content recognition within the same framework, which simplifies the evaluation and enables more meaningful comparisons across different types of TSR approaches. Code will be released at https://github.com/microsoft/table-transformer.

A recommendation system uses the past purchases or ratings of n products by a group of m users, in order to provide personalized recommendations to individual users. The information is modeled as an m times n preference matrix which is assumed to have a good rank-k approximation, for a small constant k. In this work, we present a quantum algorithm for recommendation systems that has running time O(poly(k)polylog(mn)). All known classical algorithms for recommendation systems that work through reconstructing an approximation of the preference matrix run in time polynomial in the matrix dimension. Our algorithm provides good recommendations by sampling efficiently from an approximation of the preference matrix, without reconstructing the entire matrix. For this, we design an efficient quantum procedure to project a given vector onto the row space of a given matrix. This is the first algorithm for recommendation systems that runs in time polylogarithmic in the dimensions of the matrix and provides an example of a quantum machine learning algorithm for a real world application.

Geometric alignment scores based on Gram determinants provide a compact way to model higher-order consistency among modalities, yet how such scores respond to modality degradation is poorly understood. This paper asks whether the response of a multimodal geometric score is determined primarily by the magnitude of the perturbation-induced displacement. Using frozen cohorts from MSR-VTT (N=878) and DiDeMo (N=980), we apply controlled video blur and audio noise and analyze the response in the relational geometry on which the score is defined. Displacement magnitude explains at most 15% of the out-of-sample variance in the absolute response, and magnitude-matched pairs respond systematically differently, so scalar magnitude does not organize the response. The closed-form first-order expansion of the Gramian volume yields the Directional Geometric Response (DGR): the projection of the displacement onto the local volume gradient, which jointly captures the clean operating point, displacement magnitude, and displacement direction. The absolute first-order DGR term explains the observed response with out-of-sample R^2 of 0.838-0.969, matched-magnitude ranking accuracies of 0.864-0.963, and response-sign accuracies of 0.909-0.989, whereas the tested direction-free alternatives remain weak or unstable under the corresponding evaluation protocols. A pre-specified gain-normalization candidate, V/(g_V+eps), fails its predictability and clean-order gates. DGR uses the observed degraded-state displacement and is therefore an explanatory quantity, not a deployment-time predictor: geometric response depends on where the representation operates, how far degradation moves the relational geometry, and in which direction it moves.

A new generalized matrix inverse is derived which is consistent with respect to arbitrary nonsingular diagonal transformations, e.g., it preserves units associated with variables under state space transformations, thus providing a general solution to a longstanding open problem relevant to a wide variety of applications in robotics, tracking, and control systems. The new inverse complements the Drazin inverse (which is consistent with respect to similarity transformations) and the Moore-Penrose inverse (which is consistent with respect to unitary/orthonormal transformations) to complete a trilogy of generalized matrix inverses that exhausts the standard family of analytically-important linear system transformations. Results are generalized to obtain unit-consistent and unit-invariant matrix decompositions and examples of their use are described.

Sparse linear regression is one of the most basic questions in machine learning and statistics. Here, we are given as input a design matrix X in R^{N times d} and measurements or labels {y} in R^N where {y} = {X} {w}^* + {xi}, and {xi} is the noise in the measurements. Importantly, we have the additional constraint that the unknown signal vector {w}^* is sparse: it has k non-zero entries where k is much smaller than the ambient dimension. Our goal is to output a prediction vector {w} that has small prediction error: 1{N}cdot |{X} {w}^* - {X} {w}|^2_2. Information-theoretically, we know what is best possible in terms of measurements: under most natural noise distributions, we can get prediction error at most epsilon with roughly N = O(k log d/epsilon) samples. Computationally, this currently needs d^{Omega(k)} run-time. Alternately, with N = O(d), we can get polynomial-time. Thus, there is an exponential gap (in the dependence on d) between the two and we do not know if it is possible to get d^{o(k)} run-time and o(d) samples. We give the first generic positive result for worst-case design matrices {X}: For any {X}, we show that if the support of {w}^* is chosen at random, we can get prediction error epsilon with N = poly(k, log d, 1/epsilon) samples and run-time poly(d,N). This run-time holds for any design matrix {X} with condition number up to 2^{poly(d)}. Previously, such results were known for worst-case {w}^*, but only for random design matrices from well-behaved families, matrices that have a very low condition number (poly(log d); e.g., as studied in compressed sensing), or those with special structural properties.

Modern Large Language Models (LLMs) are composed of matrices with billions of elements, making their storage and processing quite demanding in terms of computational resources and memory usage. Being significantly large, such matrices can often be expressed in low-rank format with potential to relax resource requirements. Unlike prior works which focus on developing novel matrix decomposition algorithms, in this work we first study the emergence of low-rank structures across matrices within different layers of LLMs and establish a consequential relationship between the gradient dynamics and emerging low-rank expressiveness of matrices. Our findings reveal that different layers exhibit varying levels of converged low-rank structure, necessitating a non-uniform rank reduction across them to minimize performance drop due to compression. In view of that, we present Weight Low-Rank Projection (WeLore) that unifies weight compression and memory-efficient fine-tuning as ONE, in a data-agnostic and one-shot way. WeLore capitalizes the heavy-tail distribution of singular values to identify a suitable rank reduction ratio for matrices within LLMs. Going beyond only as a compression technique, WeLore categorizes weight matrices into Low-rank Components (LRCs) and Non-Low-rank Components (N-LRCs) based on their ability to express themselves as low-rank. Our gradient perspective and extensive experiments illustrate that LRCs tend to have better finetuning capabilities and can closely mimic (sometimes outperform) the training loss trajectory and performance of full-finetuning with notable memory and compute footprint reduction. For example, finetuning a 50\% compressed LLaMa-2 7B model using only a fraction of parameters in LRCs (WeLore) can outperform its full finetuning with ~3x better throughput and ~0.6x GPU requirement. Our codes are available at https://github.com/VITA-Group/welore

We consider the problem of estimating (diagonally dominant) M-matrices as precision matrices in Gaussian graphical models. These models exhibit intriguing properties, such as the existence of the maximum likelihood estimator with merely two observations for M-matrices lauritzen2019maximum,slawski2015estimation and even one observation for diagonally dominant M-matrices truell2021maximum. We propose an adaptive multiple-stage estimation method that refines the estimate by solving a weighted ell_1-regularized problem at each stage. Furthermore, we develop a unified framework based on the gradient projection method to solve the regularized problem, incorporating distinct projections to handle the constraints of M-matrices and diagonally dominant M-matrices. A theoretical analysis of the estimation error is provided. Our method outperforms state-of-the-art methods in precision matrix estimation and graph edge identification, as evidenced by synthetic and financial time-series data sets.

This document presents a series of open questions arising in matrix computations, i.e., the numerical solution of linear algebra problems. It is a result of working groups at the workshop Linear Systems and Eigenvalue Problems, which was organized at the Simons Institute for the Theory of Computing program on Complexity and Linear Algebra in Fall 2025. The complexity and numerical solution of linear algebra problems is a crosscutting area between theoretical computer science and numerical analysis. The value of the particular problem formulations here is that they were produced via discussions between researchers from both groups. The open questions are organized in five categories: iterative solvers for linear systems, eigenvalue computation, low-rank approximation, randomized sketching, and other areas including tensors, quantum systems, and matrix functions. (Updated to reflect the status of the open problems as of August 20, 2026.)

We introduce ReALLM, a novel approach for compression and memory-efficient adaptation of pre-trained language models that encompasses most of the post-training quantization and fine-tuning methods for a budget of <4 bits. Pre-trained matrices are decomposed into a high-precision low-rank component and a vector-quantized latent representation (using an autoencoder). During the fine-tuning step, only the low-rank components are updated. Our results show that pre-trained matrices exhibit different patterns. ReALLM adapts the shape of the encoder (small/large embedding, high/low bit VQ, etc.) to each matrix. ReALLM proposes to represent each matrix with a small embedding on b bits and a neural decoder model D_phi with its weights on b_phi bits. The decompression of a matrix requires only one embedding and a single forward pass with the decoder. Our weight-only quantization algorithm yields the best results on language generation tasks (C4 and WikiText-2) for a budget of 3 bits without any training. With a budget of 2 bits, ReALLM achieves state-of-the art performance after fine-tuning on a small calibration dataset.

Matrix manifolds, such as manifolds of Symmetric Positive Definite (SPD) matrices and Grassmann manifolds, appear in many applications. Recently, by applying the theory of gyrogroups and gyrovector spaces that is a powerful framework for studying hyperbolic geometry, some works have attempted to build principled generalizations of Euclidean neural networks on matrix manifolds. However, due to the lack of many concepts in gyrovector spaces for the considered manifolds, e.g., the inner product and gyroangles, techniques and mathematical tools provided by these works are still limited compared to those developed for studying hyperbolic geometry. In this paper, we generalize some notions in gyrovector spaces for SPD and Grassmann manifolds, and propose new models and layers for building neural networks on these manifolds. We show the effectiveness of our approach in two applications, i.e., human action recognition and knowledge graph completion.

We study the complexity of sampling from a distribution over all index subsets of the set {1,...,n} with the probability of a subset S proportional to the determinant of the submatrix L_S of some ntimes n p.s.d. matrix L, where L_S corresponds to the entries of L indexed by S. Known as a determinantal point process, this distribution is used in machine learning to induce diversity in subset selection. In practice, we often wish to sample multiple subsets S with small expected size k = E[|S|] ll n from a very large matrix L, so it is important to minimize the preprocessing cost of the procedure (performed once) as well as the sampling cost (performed repeatedly). For this purpose, we propose a new algorithm which, given access to L, samples exactly from a determinantal point process while satisfying the following two properties: (1) its preprocessing cost is n cdot poly(k), i.e., sublinear in the size of L, and (2) its sampling cost is poly(k), i.e., independent of the size of L. Prior to our results, state-of-the-art exact samplers required O(n^3) preprocessing time and sampling time linear in n or dependent on the spectral properties of L. We also give a reduction which allows using our algorithm for exact sampling from cardinality constrained determinantal point processes with ncdotpoly(k) time preprocessing.

We revisit the well-studied problem of approximating a matrix product, A^TB, based on small space sketches S(A) and S(B) of A in R^{n times d} and Bin R^{n times m}. We are interested in the setting where the sketches must be computed independently of each other, except for the use of a shared random seed. We prove that, when A and B are sparse, methods based on coordinated random sampling can outperform classical linear sketching approaches, like Johnson-Lindenstrauss Projection or CountSketch. For example, to obtain Frobenius norm error epsilon|A|_F|B|_F, coordinated sampling requires sketches of size O(s/epsilon^2) when A and B have at most s leq d,m non-zeros per row. In contrast, linear sketching leads to sketches of size O(d/epsilon^2) and O(m/epsilon^2) for A and B. We empirically evaluate our approach on two applications: 1) distributed linear regression in databases, a problem motivated by tasks like dataset discovery and augmentation, and 2) approximating attention matrices in transformer-based language models. In both cases, our sampling algorithms yield an order of magnitude improvement over linear sketching.

The Gram-Schmidt Walk is a randomized vector-balancing algorithm whose subgaussian guarantees support applications in discrepancy, experimental design, and data compression; however, these theoretical guarantees are established in exact arithmetic, whereas implementations must approximate least-squares directions, boundary updates, and sampling probabilities in finite precision. This is important because small numerical errors can change which coordinates freeze and thereby alter the subsequent trajectory. We analyze the concentration of the perturbed Gram-Schmidt walk directly under bounded, potentially biased and history-dependent errors. For n input vectors of Euclidean norm at most one, we obtain a modified MGF bound depending on key error sources which recovers the original result as the error goes to zero. We also construct a full-column-rank instance in which bounded update errors produce bias of order min{n^2varepsilon,n}, showing that updates accumulate error unavoidably under this model. Finally, we validate our findings in a variety of settings by ablating on the bit precision and problem size.

Given a matrix Ain R^{ntimes d} and a vector bin R^n, we consider the regression problem with ell_infty guarantees: finding a vector x'in R^d such that |x'-x^*|_infty leq epsilon{d}cdot |Ax^*-b|_2cdot |A^dagger| where x^*=argmin_{xin R^d}|Ax-b|_2. One popular approach for solving such ell_2 regression problem is via sketching: picking a structured random matrix Sin R^{mtimes n} with mll n and SA can be quickly computed, solve the ``sketched'' regression problem argmin_{xin R^d} |SAx-Sb|_2. In this paper, we show that in order to obtain such ell_infty guarantee for ell_2 regression, one has to use sketching matrices that are dense. To the best of our knowledge, this is the first user case in which dense sketching matrices are necessary. On the algorithmic side, we prove that there exists a distribution of dense sketching matrices with m=epsilon^{-2}dlog^3(n/delta) such that solving the sketched regression problem gives the ell_infty guarantee, with probability at least 1-delta. Moreover, the matrix SA can be computed in time O(ndlog n). Our row count is nearly-optimal up to logarithmic factors, and significantly improves the result in [Price, Song and Woodruff, ICALP'17], in which a super-linear in d rows, m=Omega(epsilon^{-2}d^{1+gamma}) for gamma=Theta(frac{loglog n{log d}}) is required. We also develop a novel analytical framework for ell_infty guarantee regression that utilizes the Oblivious Coordinate-wise Embedding (OCE) property introduced in [Song and Yu, ICML'21]. Our analysis is arguably much simpler and more general than [Price, Song and Woodruff, ICALP'17], and it extends to dense sketches for tensor product of vectors.

Are n-gram language models still relevant in this era of neural large language models (LLMs)? Our answer is yes, and we show their values in both text analysis and improving neural LLMs. Yet this necessitates modernizing n-gram models in two aspects. First, we train them at the same data scale as neural LLMs -- 1.4 trillion tokens. This is the largest n-gram model ever built. Second, existing n-gram models use small n which hinders their performance; we instead allow n to be arbitrarily large, by introducing a new infty-gram LM with backoff. Instead of pre-computing n-gram count tables (which would be very expensive), we develop an engine named infini-gram -- powered by suffix arrays -- that can compute infty-gram (as well as n-gram with arbitrary n) probabilities with millisecond-level latency. The infty-gram framework and infini-gram engine enable us to conduct many novel and interesting analyses of human-written and machine-generated text: we find that the infty-gram LM has fairly high accuracy for next-token prediction (47%), and can complement neural LLMs to greatly reduce their language modeling perplexities. When analyzing machine-generated text, we also observe irregularities in the machine--infty-gram agreement level with respect to the suffix length, which indicates deficiencies in neural LLM pretraining and the positional embeddings of Transformers. We open-source our infini-gram engine in the hopes of enabling more study on how to best use verbatim information retrieved from large text corpora.

Motivated by the problem of fast processing of attention matrices, we study fast algorithms for computing matrix-vector products for asymmetric Gaussian Kernel matrices Kin R^{ntimes n}. K's columns are indexed by a set of n keys k_1,k_2ldots, k_nin R^d, rows by a set of n queries q_1,q_2,ldots,q_nin R^d , and its i,j entry is K_{ij} = e^{-|q_i-k_j|_2^2/2sigma^2} for some bandwidth parameter sigma>0. Given a vector xin R^n and error parameter epsilon>0, our task is to output a yin R^n such that |Kx-y|_2leq epsilon |x|_2 in time subquadratic in n and linear in d. Our algorithms rely on the following modelling assumption about the matrices K: the sum of the entries of K scales linearly in n, as opposed to worst case quadratic growth. We validate this assumption experimentally, for Gaussian kernel matrices encountered in various settings such as fast attention computation in LLMs. We obtain the first subquadratic-time algorithm that works under this assumption, for unrestricted vectors.

Recently, there has been a surge of interest in employing neural networks for graph generation, a fundamental statistical learning problem with critical applications like molecule design and community analysis. However, most approaches encounter significant limitations when generating large-scale graphs. This is due to their requirement to output the full adjacency matrices whose size grows quadratically with the number of nodes. In response to this challenge, we introduce a new, simple, and scalable graph representation named gap encoded edge list (GEEL) that has a small representation size that aligns with the number of edges. In addition, GEEL significantly reduces the vocabulary size by incorporating the gap encoding and bandwidth restriction schemes. GEEL can be autoregressively generated with the incorporation of node positional encoding, and we further extend GEEL to deal with attributed graphs by designing a new grammar. Our findings reveal that the adoption of this compact representation not only enhances scalability but also bolsters performance by simplifying the graph generation process. We conduct a comprehensive evaluation across ten non-attributed and two molecular graph generation tasks, demonstrating the effectiveness of GEEL.

In recent years, random matrices have come to play a major role in computational mathematics, but most of the classical areas of random matrix theory remain the province of experts. Over the last decade, with the advent of matrix concentration inequalities, research has advanced to the point where we can conquer many (formerly) challenging problems with a page or two of arithmetic. The aim of this monograph is to describe the most successful methods from this area along with some interesting examples that these techniques can illuminate.

We consider in this work quantities that can be obtained as limits of powers of parametrized matrices, for instance the inverse matrix or the logarithm of the determinant. Under the assumption of affine dependence in the parameters, we use the Empirical Interpolation Method (EIM) to derive an approximation for powers of these matrices, from which we derive a nonintrusive approximation for the aforementioned limits. We derive upper bounds of the error made by the obtained formula. Finally, numerical comparisons with classical intrusive and nonintrusive approximation techniques are provided: in the considered test-cases, our algorithm performs well compared to the nonintrusive ones.

Deep subspace clustering (DSC) networks based on self-expressive model learn representation matrix, often implemented in terms of fully connected network, in the embedded space. After the learning is finished, representation matrix is used by spectral clustering module to assign labels to clusters. However, such approach ignores complementary information that exist in other layers of the encoder (including the input data themselves). Herein, we apply selected linear subspace clustering algorithm to learn representation matrices from representations learned by all layers of encoder network including the input data. Afterward, we learn a multilayer graph that in a multi-view like manner integrates information from graph Laplacians of all used layers. That improves further performance of selected DSC network. Furthermore, we also provide formulation of our approach to cluster out-of-sample/test data points. We validate proposed approach on four well-known datasets with two DSC networks as baseline models. In almost all the cases, proposed approach achieved statistically significant improvement in three performance metrics. MATLAB code of proposed algorithm is posted on https://github.com/lovro-sinda/MLG-DSC.

With the recent emergence of mixed precision hardware, there has been a renewed interest in its use for solving numerical linear algebra problems fast and accurately. The solution of least squares (LS) problems min_x|b-Ax|_2, where A in R^{mtimes n}, arise in numerous application areas. Overdetermined standard least squares problems can be solved by using mixed precision within the iterative refinement method of Björck, which transforms the least squares problem into an (m+n)times(m+n) ''augmented'' system. It has recently been shown that mixed precision GMRES-based iterative refinement can also be used, in an approach termed GMRES-LSIR. In practice, we often encounter types of least squares problems beyond standard least squares, including weighted least squares (WLS), min_x|D^{1/2}(b-Ax)|_2, where D^{1/2} is a diagonal matrix of weights. In this paper, we discuss a mixed precision FGMRES-WLSIR algorithm for solving WLS problems using two different preconditioners.

Projection maintenance is one of the core data structure tasks. Efficient data structures for projection maintenance have led to recent breakthroughs in many convex programming algorithms. In this work, we further extend this framework to the Kronecker product structure. Given a constraint matrix {sf A} and a positive semi-definite matrix Win R^{ntimes n} with a sparse eigenbasis, we consider the task of maintaining the projection in the form of {sf B}^top({sf B}{sf B}^top)^{-1}{sf B}, where {sf B}={sf A}(Wotimes I) or {sf B}={sf A}(W^{1/2}otimes W^{1/2}). At each iteration, the weight matrix W receives a low rank change and we receive a new vector h. The goal is to maintain the projection matrix and answer the query {sf B}^top({sf B}{sf B}^top)^{-1}{sf B}h with good approximation guarantees. We design a fast dynamic data structure for this task and it is robust against an adaptive adversary. Following the beautiful and pioneering work of [Beimel, Kaplan, Mansour, Nissim, Saranurak and Stemmer, STOC'22], we use tools from differential privacy to reduce the randomness required by the data structure and further improve the running time.

We give explicit low-rank bilinear non-commutative schemes for multiplying structured n times n matrices with 2 leq n leq 5, which serve as building blocks for recursive algorithms with improved multiplicative factors in asymptotic complexity. Our schemes are discovered over F_2 or F_3 and lifted to Z or Q. Using a flip graph search over tensor decompositions, we derive schemes for general, upper-triangular, lower-triangular, symmetric, and skew-symmetric inputs, as well as products of a structured matrix with its transpose. In particular, we obtain 4 times 4 rank-34 schemes: (i) multiplying a general matrix by its transpose using 10 recursive calls, improving the factor from 26/41 (0.634) to 8/13 (0.615); and (ii) multiplying an upper-triangular matrix by a general matrix using 12 recursive calls, improving the factor from 8/13 (0.615) to 22/37 (0.595). Additionally, using F_3 flip graphs, we discover schemes over Q that fundamentally require the inverse of 2, including a 2 times 2 symmetric-symmetric multiplication of rank 5 and a 3 times 3 skew-symmetric-general multiplication of rank 14 (improving upon AlphaTensor's 15).

Low-rank optimization has emerged as a promising direction in training large language models (LLMs) to reduce the memory usage of adaptive optimizers by constraining learning to a lower-dimensional space. Prior work typically projects gradients of linear layers using approaches based on Singular Value Decomposition (SVD). However, applying SVD-based procedures individually to each layer in large models is computationally expensive and incurs additional memory costs due to storing the projection matrices. In this work, we propose a computationally efficient and conceptually simple two-step procedure to approximate SVD-based gradient projections into lower-dimensional spaces. First, we construct a complete orthogonal basis using predefined orthogonal matrices of the Discrete Cosine Transform (DCT). Second, we adaptively select basis columns based on their alignment with the gradient of each layer. Each projection matrix in our method is obtained via a single matrix multiplication followed by a lightweight sorting step to identify the most relevant basis vectors. Due to the predefined nature of the orthogonal bases, they are computed once at the start of training. During training, we store only the indices of the selected columns, avoiding the need to store full projection matrices for each layer. Our numerical experiments on both pre-training and fine-tuning tasks demonstrate the effectiveness of our dual strategy in approximating optimal low-rank projections, matching the performance of costly SVD-based methods while achieving faster runtime and reduced memory usage.

Since the control of the Lipschitz constant has a great impact on the training stability, generalization, and robustness of neural networks, the estimation of this value is nowadays a real scientific challenge. In this paper we introduce a precise, fast, and differentiable upper bound for the spectral norm of convolutional layers using circulant matrix theory and a new alternative to the Power iteration. Called the Gram iteration, our approach exhibits a superlinear convergence. First, we show through a comprehensive set of experiments that our approach outperforms other state-of-the-art methods in terms of precision, computational cost, and scalability. Then, it proves highly effective for the Lipschitz regularization of convolutional neural networks, with competitive results against concurrent approaches. Code is available at https://github.com/blaisedelattre/lip4conv.

We introduce meta-factorization, a theory that describes matrix decompositions as solutions of linear matrix equations: the projector and the reconstruction equation. Meta-factorization reconstructs known factorizations, reveals their internal structures, and allows for introducing modifications, as illustrated with SVD, QR, and UTV factorizations. The prospect of meta-factorization also provides insights into computational aspects of generalized matrix inverses and randomized linear algebra algorithms. The relations between the Moore-Penrose pseudoinverse, generalized Nystr\"{o}m method, and the CUR decomposition are revealed here as an illustration. Finally, meta-factorization offers hints on the structure of new factorizations and provides the potential of creating them.

Despite the widespread empirical success of ResNet, the generalization properties of deep ResNet are rarely explored beyond the lazy training regime. In this work, we investigate scaled ResNet in the limit of infinitely deep and wide neural networks, of which the gradient flow is described by a partial differential equation in the large-neural network limit, i.e., the mean-field regime. To derive the generalization bounds under this setting, our analysis necessitates a shift from the conventional time-invariant Gram matrix employed in the lazy training regime to a time-variant, distribution-dependent version. To this end, we provide a global lower bound on the minimum eigenvalue of the Gram matrix under the mean-field regime. Besides, for the traceability of the dynamic of Kullback-Leibler (KL) divergence, we establish the linear convergence of the empirical error and estimate the upper bound of the KL divergence over parameters distribution. Finally, we build the uniform convergence for generalization bound via Rademacher complexity. Our results offer new insights into the generalization ability of deep ResNet beyond the lazy training regime and contribute to advancing the understanding of the fundamental properties of deep neural networks.

Deepseek's Engram, a conditional memory module, was introduced to trade-off storage versus reasoning in large language models. However, the module relies on token-level N-gram hashing for Engram embedding lookup, introducing a tight coupling to the tokenizer used: a model with a different tokenizer would have to train its own Engram embeddings from scratch. To improve the reusability of Engram embeddings, we propose a change to the hashing routine, enabling compatibility between Engram models using different tokenizers. Instead of modelling disjoint N-gram spaces, we treat N-gram as a method to sample potentially useful byte sequences, from all possible byte sequences across tokens. We replace the XOR-based hashing with the general polynomial hashing with a joint embedding space across N. This work investigates the possible trade-offs and shows that this simple substitution produces comparable performance and achieves tokenizer-agnosticism: hash equivalence for byte-equivalent token sequences.

As large language models (LLMs) continue to evolve, efficient evaluation metrics are vital for assessing their ability to compress information and reduce redundancy. While traditional metrics like Matrix Entropy offer valuable insights, they are computationally intensive for large-scale models due to their \( O(n^3) \) time complexity with Singular Value Decomposition (SVD). To mitigate this issue, we introduce the Matrix Nuclear-Norm, which not only serves as a metric to quantify the data compression proficiency of LLM but also provides a convex approximation of matrix rank to capture both predictive discriminability and diversity. By employing the \( L_{1,2}-norm \) to further approximate the nuclear norm, we can effectively assess the model's information compression capabilities. This approach reduces the time complexity to \( O(n^2) \) and eliminates the need for SVD computation. Consequently, the Matrix Nuclear-Norm achieves speeds 8 to 24 times faster than Matrix Entropy for the CEREBRAS-GPT model as sizes increase from 111M to 6.7B. This performance gap becomes more pronounced with larger models, as validated in tests with other models like Pythia. Additionally, evaluations on benchmarks and model responses confirm that our proposed Matrix Nuclear-Norm is a reliable, scalable, and efficient tool for assessing LLMs' performance, striking a balance between accuracy and computational efficiency. The code is available at https://github.com/MLGroupJLU/MatrixNuclearNorm.

Riemannian geometry has been applied to Brain Computer Interface (BCI) for brain signals classification yielding promising results. Studying electroencephalographic (EEG) signals from their associated covariance matrices allows a mitigation of common sources of variability (electronic, electrical, biological) by constructing a representation which is invariant to these perturbations. While working in Euclidean space with covariance matrices is known to be error-prone, one might take advantage of algorithmic advances in information geometry and matrix manifold to implement methods for Symmetric Positive-Definite (SPD) matrices. This paper proposes a comprehensive review of the actual tools of information geometry and how they could be applied on covariance matrices of EEG. In practice, covariance matrices should be estimated, thus a thorough study of all estimators is conducted on real EEG dataset. As a main contribution, this paper proposes an online implementation of a classifier in the Riemannian space and its subsequent assessment in Steady-State Visually Evoked Potential (SSVEP) experimentations.

The learnable, linear neural network layers between tensor power spaces of R^{n} that are equivariant to the orthogonal group, O(n), the special orthogonal group, SO(n), and the symplectic group, Sp(n), were characterised in arXiv:2212.08630. We present an algorithm for multiplying a vector by any weight matrix for each of these groups, using category theoretic constructions to implement the procedure. We achieve a significant reduction in computational cost compared with a naive implementation by making use of Kronecker product matrices to perform the multiplication. We show that our approach extends to the symmetric group, S_n, recovering the algorithm of arXiv:2303.06208 in the process.

Continuous word representations, trained on large unlabeled corpora are useful for many natural language processing tasks. Popular models that learn such representations ignore the morphology of words, by assigning a distinct vector to each word. This is a limitation, especially for languages with large vocabularies and many rare words. In this paper, we propose a new approach based on the skipgram model, where each word is represented as a bag of character n-grams. A vector representation is associated to each character n-gram; words being represented as the sum of these representations. Our method is fast, allowing to train models on large corpora quickly and allows us to compute word representations for words that did not appear in the training data. We evaluate our word representations on nine different languages, both on word similarity and analogy tasks. By comparing to recently proposed morphological word representations, we show that our vectors achieve state-of-the-art performance on these tasks.

Dictionary learning is a branch of signal processing and machine learning that aims at finding a frame (called dictionary) in which some training data admits a sparse representation. The sparser the representation, the better the dictionary. The resulting dictionary is in general a dense matrix, and its manipulation can be computationally costly both at the learning stage and later in the usage of this dictionary, for tasks such as sparse coding. Dictionary learning is thus limited to relatively small-scale problems. In this paper, inspired by usual fast transforms, we consider a general dictionary structure that allows cheaper manipulation, and propose an algorithm to learn such dictionaries --and their fast implementation-- over training data. The approach is demonstrated experimentally with the factorization of the Hadamard matrix and with synthetic dictionary learning experiments.

Large language model (LLM) decoding involves generating a sequence of tokens based on a given context, where each token is predicted one at a time using the model's learned probabilities. The typical autoregressive decoding method requires a separate forward pass through the model for each token generated, which is computationally inefficient and poses challenges for deploying LLMs in latency-sensitive scenarios. The main limitations of current decoding methods stem from their inefficiencies and resource demands. Existing approaches either necessitate fine-tuning smaller models, which is resource-intensive, or rely on fixed retrieval schemes to construct drafts for the next tokens, which lack adaptability and fail to generalize across different models and contexts. To address these issues, we introduce a novel methodology called ADED, which accelerates LLM decoding without requiring fine-tuning. Our approach involves an adaptive draft-verification process that evolves over time to improve efficiency. We utilize a tri-gram matrix-based LLM representation to dynamically approximate the output distribution of the LLM, allowing the model to adjust to changing token probabilities during the decoding process. Additionally, we implement a draft construction mechanism that effectively balances exploration and exploitation, ensuring that the drafts generated are both diverse and close to the true output distribution of the LLM. The importance of this design lies in its ability to optimize the draft distribution adaptively, leading to faster and more accurate decoding. Through extensive experiments on various benchmark datasets and LLM architectures, we demonstrate that ADED significantly accelerates the decoding process while maintaining high accuracy, making it suitable for deployment in a wide range of practical applications.

We study the matrix models pi:C(S_N^+)to M_N(C(X)) which are flat, in the sense that the standard generators of C(S_N^+) are mapped to rank 1 projections. Our first result is a generalization of the Pauli matrix construction at N=4, using finite groups and 2-cocycles. Our second result is the construction of a universal representation of C(S_N^+), inspired from the Sinkhorn algorithm, that we conjecture to be inner faithful.

Linear Recurrent Neural Networks (linear RNNs) have emerged as competitive alternatives to Transformers for sequence modeling, offering efficient training and linear-time inference. However, existing architectures face a fundamental trade-off between expressivity and efficiency, dictated by the structure of their state-transition matrices. Diagonal matrices, used in models such as Mamba, GLA, or mLSTM, yield fast runtime but have limited expressivity. To address this, recent architectures such as DeltaNet and RWKV-7 adopted a diagonal plus rank-1 structure, which allows simultaneous token and channel mixing, improving associative recall and, as recently shown, state-tracking when allowing negative eigenvalues in the state-transition matrices. Building on the interpretation of DeltaNet's recurrence as performing one step of online gradient descent per token on an associative recall loss, we introduce DeltaProduct, which instead takes multiple (n_h) steps per token. This naturally leads to diagonal plus rank-n_h state-transition matrices, formed as products of n_h generalized Householder transformations, providing a tunable mechanism to balance expressivity and efficiency. We provide a detailed theoretical characterization of the state-tracking capability of DeltaProduct in finite precision, showing how it improves by increasing n_h. Our extensive experiments demonstrate that DeltaProduct outperforms DeltaNet in both state-tracking and language modeling, while also showing significantly improved length extrapolation capabilities.

Daniel McNulty et.al (2024 J. Phys. A: Math. Theor. submitted) voiced suspicions to the Lemma 11(v) Part 6 in Chen and Yu (2017 J. Phys. A: Math. Theor. 50 475304) and three theorems derived in later publications (Liang et al 2019 Quantum Inf. Process. 18 352; Liang et al 2019 Linear Multilinear Algebra 69 2908-25; Chen et al 2021 Quantum Inf. Process. 20 353). For these suspicions, we reprove that any complex Hadamard matrix of order-six whose number of real elements more than 22 does not belong to a set of four mutually unbiased bases. We show that the number of complex Hadamard submatrices of order-two of any H2-reducible matrix is not 10,...,16,18. We also put forward some possible directions for further development.

We propose a new benchmark corpus to be used for measuring progress in statistical language modeling. With almost one billion words of training data, we hope this benchmark will be useful to quickly evaluate novel language modeling techniques, and to compare their contribution when combined with other advanced techniques. We show performance of several well-known types of language models, with the best results achieved with a recurrent neural network based language model. The baseline unpruned Kneser-Ney 5-gram model achieves perplexity 67.6; a combination of techniques leads to 35% reduction in perplexity, or 10% reduction in cross-entropy (bits), over that baseline. The benchmark is available as a code.google.com project; besides the scripts needed to rebuild the training/held-out data, it also makes available log-probability values for each word in each of ten held-out data sets, for each of the baseline n-gram models.

Ensembles of neural network weight matrices are studied through the training process for the MNIST classification problem, testing the efficacy of matrix models for representing their distributions, under assumptions of Gaussianity and permutation-symmetry. The general 13-parameter permutation invariant Gaussian matrix models are found to be effective models for the correlated Gaussianity in the weight matrices, beyond the range of applicability of the simple Gaussian with independent identically distributed matrix variables, and notably well beyond the initialisation step. The representation theoretic model parameters, and the graph-theoretic characterisation of the permutation invariant matrix observables give an interpretable framework for the best-fit model and for small departures from Gaussianity. Additionally, the Wasserstein distance is calculated for this class of models and used to quantify the movement of the distributions over training. Throughout the work, the effects of varied initialisation regimes, regularisation, layer depth, and layer width are tested for this formalism, identifying limits where particular departures from Gaussianity are enhanced and how more general, yet still highly-interpretable, models can be developed.

The Land Matrix initiative (https://landmatrix.org) and its global observatory aim to provide reliable data on large-scale land acquisitions to inform debates and actions in sectors such as agriculture, extraction, or energy in low- and middle-income countries. Although these data are recognized in the academic world, they remain underutilized in public policy, mainly due to the complexity of access and exploitation, which requires technical expertise and a good understanding of the database schema. The objective of this work is to simplify access to data from different database systems. The methods proposed in this article are evaluated using data from the Land Matrix. This work presents various comparisons of Large Language Models (LLMs) as well as combinations of LLM adaptations (Prompt Engineering, RAG, Agents) to query different database systems (GraphQL and REST queries). The experiments are reproducible, and a demonstration is available online: https://github.com/tetis-nlp/landmatrix-graphql-python.

By interpreting the product of the Principal Component Analysis, that is the covariance matrix, as a sequence of nested subspaces naturally coming with weights according to the level of approximation they provide, we are able to embed all d--dimensional Grassmannians into a stratified space of covariance matrices. We observe that Grassmannians constitute the lowest dimensional skeleton of the stratification while it is possible to define a Riemaniann metric on the highest dimensional and dense stratum, such a metric being compatible with the global stratification. With such a Riemaniann metric at hand, it is possible to look for geodesics between two linear subspaces of different dimensions that do not go through higher dimensional linear subspaces as would euclidean geodesics. Building upon the proposed embedding of Grassmannians into the stratified space of covariance matrices, we generalize the concept of varifolds to what we call flagfolds in order to model multi-dimensional shapes.

Let G be a graph with adjacency matrix A(G) and degree matrix D(G), and let L_μ(G):=A(G)-μD(G). Two graphs G_1 and G_2 are called degree-similar if there exists an invertible matrix M such that M^{-1} A(G_1) M =A(G_2) and M^{-1} D(G_1) M =D(G_2). In this paper, we address three problems concerning degree-similar graphs proposed by Godsil and Sun. First, we present a new characterization of degree-similar graphs using degree partition, from which we derive methods and examples for constructing cospectral graphs and degree-similar graphs. Second, we construct infinite pairs of non-degree-similar trees G_1 and G_2 such that tI- L_μ(G_1) and tI-L_μ(G_2) have the same Smith normal form over Q(μ)[t], which provides a negative answer to a problem posed by Godsil and Sun. Third, we establish several invariants of degree-similar graphs and obtain results on unicyclic graphs that are degree-similar determined. Lastly we prove that for a strongly regular graph G and any two edges e and f of G, G backslash e and G backslash f have identical μ-polynomial, i.e., det(tI-L_μ(G backslash e))=det(tI-L_μ(G backslash f)), which enables the construction of pairs of non-isomorphic graphs with same μ-polynomial, where G backslash e denotes the graph obtained from G by deleting the edge e.

Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b, find a vector x such that Ax=b. We consider the case where one doesn't need to know the solution x itself, but rather an approximation of the expectation value of some operator associated with x, e.g., x'Mx for some matrix M. In this case, when A is sparse, N by N and has condition number kappa, classical algorithms can find x and estimate x'Mx in O(N sqrt(kappa)) time. Here, we exhibit a quantum algorithm for this task that runs in poly(log N, kappa) time, an exponential improvement over the best classical algorithm.

We introduce ChatGLM, an evolving family of large language models that we have been developing over time. This report primarily focuses on the GLM-4 language series, which includes GLM-4, GLM-4-Air, and GLM-4-9B. They represent our most capable models that are trained with all the insights and lessons gained from the preceding three generations of ChatGLM. To date, the GLM-4 models are pre-trained on ten trillions of tokens mostly in Chinese and English, along with a small set of corpus from 24 languages, and aligned primarily for Chinese and English usage. The high-quality alignment is achieved via a multi-stage post-training process, which involves supervised fine-tuning and learning from human feedback. Evaluations show that GLM-4 1) closely rivals or outperforms GPT-4 in terms of general metrics such as MMLU, GSM8K, MATH, BBH, GPQA, and HumanEval, 2) gets close to GPT-4-Turbo in instruction following as measured by IFEval, 3) matches GPT-4 Turbo (128K) and Claude 3 for long context tasks, and 4) outperforms GPT-4 in Chinese alignments as measured by AlignBench. The GLM-4 All Tools model is further aligned to understand user intent and autonomously decide when and which tool(s) touse -- including web browser, Python interpreter, text-to-image model, and user-defined functions -- to effectively complete complex tasks. In practical applications, it matches and even surpasses GPT-4 All Tools in tasks like accessing online information via web browsing and solving math problems using Python interpreter. Over the course, we have open-sourced a series of models, including ChatGLM-6B (three generations), GLM-4-9B (128K, 1M), GLM-4V-9B, WebGLM, and CodeGeeX, attracting over 10 million downloads on Hugging face in the year 2023 alone. The open models can be accessed through https://github.com/THUDM and https://huggingface.co/THUDM.

Generating 2-by-2 unitary matrices in floating-precision arithmetic is a delicate task. One way to reduce the accumulation error is to use less floating-point operations to compute each of the entries in the 2-by-2 unitary matrix. This paper shows an algorithm that reduces the number of operations to compute the entries of a Givens rotation. Overall, the new algorithm has more operations in total when compared to algorithms in different releases of LAPACK, but less operations per entry. Numerical tests show that the new algorithm is more accurate on average.

Large language models (LLMs) are having transformative impacts across a wide range of scientific fields, particularly in the biomedical sciences. Just as the goal of Natural Language Processing is to understand sequences of words, a major objective in biology is to understand biological sequences. Genomic Language Models (gLMs), which are LLMs trained on DNA sequences, have the potential to significantly advance our understanding of genomes and how DNA elements at various scales interact to give rise to complex functions. To showcase this potential, we highlight key applications of gLMs, including functional constraint prediction, sequence design, and transfer learning. Despite notable recent progress, however, developing effective and efficient gLMs presents numerous challenges, especially for species with large, complex genomes. Here, we discuss major considerations for developing and evaluating gLMs.

Neural operator models trained on simulation data often lose accuracy when applied to experimental measurements due to the sim-to-real gap. Standard fine-tuning with limited real data can reduce this gap, but it may also damage the core physics-relevant representations learned during pretraining. Although knowledge-preserving adaptation has been widely investigated in vision or language tasks, it remains unclear whether these methods are suitable for neural operators whose architectures and protected knowledge are fundamentally different. Neural operators need to preserve core-scale physical structures rather than semantic or visual features. We propose PhysGuard, a physics-preserving framework for accurate sim-to-real adaptation of neural operators. Specifically, PhysGuard uses the empirical Fisher Information Matrix computed on simulation data to identify physics-critical parameter directions, then restricts fine-tuning updates to directions that do not interfere with them. A layer-wise Gram-matrix formulation makes this efficient for models with millions of parameters, while an adaptive threshold automatically determines the protected subspace size. A spectral probe experiment shows that the dominant Fisher directions are strongly associated with low-frequency output structures. Experiments on benchmark across four neural operator architectures and different physical systems show that PhysGuard performs strongly on most evaluation metrics compared to baselines. The benefits are most evident under severe domain shift, where it reduces low-frequency error by up to 32\% compared to standard fine-tuning while maintaining adaptability. Our code is available at https://github.com/ZhouChaunge/PhysGuard.

We study sampling algorithms for β-ensembles with time complexity less than cubic in the cardinality of the ensemble. Following Dumitriu & Edelman (2002), we see the ensemble as the eigenvalues of a random tridiagonal matrix, namely a random Jacobi matrix. First, we provide a unifying and elementary treatment of the tridiagonal models associated to the three classical Hermite, Laguerre and Jacobi ensembles. For this purpose, we use simple changes of variables between successive reparametrizations of the coefficients defining the tridiagonal matrix. Second, we derive an approximate sampler for the simulation of β-ensembles, and illustrate how fast it can be for polynomial potentials. This method combines a Gibbs sampler on Jacobi matrices and the diagonalization of these matrices. In practice, even for large ensembles, only a few Gibbs passes suffice for the marginal distribution of the eigenvalues to fit the expected theoretical distribution. When the conditionals in the Gibbs sampler can be simulated exactly, the same fast empirical convergence is observed for the fluctuations of the largest eigenvalue. Our experimental results support a conjecture by Krishnapur et al. (2016), that the Gibbs chain on Jacobi matrices of size N mixes in O(log(N)).

This paper explores the relationship between matrix factorizations and linear matrix equations. It shows that every matrix factorization defines two hidden projectors, one for the column space and one for the row space of a matrix, and how to calculate them. The projectors can be applied to solve linear matrix equations, generate low-rank approximations, or design randomized matrix algorithms. But also, as demonstrated, they can be applied in cryptography to encrypt and decrypt messages. The paper discusses some of the security implications of this application and leaves some questions open for further investigation. The basic concepts are illustrated with source code listings. Finally, this work shares some personal reflections on the meaning and importance of understanding in the time of the artificial intelligence revolution.

Coecke, Sadrzadeh, and Clark (arXiv:1003.4394v1 [cs.CL]) developed a compositional model of meaning for distributional semantics, in which each word in a sentence has a meaning vector and the distributional meaning of the sentence is a function of the tensor products of the word vectors. Abstractly speaking, this function is the morphism corresponding to the grammatical structure of the sentence in the category of finite dimensional vector spaces. In this paper, we provide a concrete method for implementing this linear meaning map, by constructing a corpus-based vector space for the type of sentence. Our construction method is based on structured vector spaces whereby meaning vectors of all sentences, regardless of their grammatical structure, live in the same vector space. Our proposed sentence space is the tensor product of two noun spaces, in which the basis vectors are pairs of words each augmented with a grammatical role. This enables us to compare meanings of sentences by simply taking the inner product of their vectors.

This paper presents several notable properties of the matrix U shown to be related to the isomorphism between H_4 and E_8. The most significant of these properties is that U.U is to rank 8 matrices what the golden ratio is to numbers. That is to say, the difference between it and its inverse is the identity element, albeit with a twist. Specifically, U.U- (U.U)^{-1} is the reverse identity matrix or standard involutory permutation matrix of rank 8. It has the same palindromic characteristic polynomial coefficients as the normalized 3-qubit Hadamard matrix with 8-bit binary basis states, which is known to be isomorphic to E8 through its (8,4) Hamming code.

Ethiopic/Amharic script is one of the oldest African writing systems, which serves at least 23 languages (e.g., Amharic, Tigrinya) in East Africa for more than 120 million people. The Amharic writing system, Abugida, has 282 syllables, 15 punctuation marks, and 20 numerals. The Amharic syllabic matrix is derived from 34 base graphemes/consonants by adding up to 12 appropriate diacritics or vocalic markers to the characters. The syllables with a common consonant or vocalic markers are likely to be visually similar and challenge text recognition tasks. In this work, we presented the first comprehensive public datasets named HUST-ART, HUST-AST, ABE, and Tana for Amharic script detection and recognition in the natural scene. We have also conducted extensive experiments to evaluate the performance of the state of art methods in detecting and recognizing Amharic scene text on our datasets. The evaluation results demonstrate the robustness of our datasets for benchmarking and its potential of promoting the development of robust Amharic script detection and recognition algorithms. Consequently, the outcome will benefit people in East Africa, including diplomats from several countries and international communities.

Short-recurrence Krylov methods that are equivalent in exact arithmetic often diverge in floating-point arithmetic. To illustrate this, we provide an exactly representable two-cycle for steepest descent with a recursively updated residual. A complementary convergence theorem gives a sufficient condition under which the stored residual decreases geometrically. We then show that a second positive definite family yields different outcomes for a Hestenes--Stiefel Conjugate Gradient (CG) implementation and a direct Lanczos--Galerkin approach. CG finds the exact solution after four updates, whereas the two-step projected system becomes inconsistent. Under stated perturbation and transfer hypotheses, a common spectral enclosure gives comparable convergence bounds. For a dense, left-to-right evaluation order, we prove a sufficient precision bound for a prescribed backward error within n updates for an ntimes n matrix, together with a computable stopping test and specified exponent-range assumptions. We use polynomial estimates and numerical experiments to examine the trade-off between working precision and the number of CG iterations needed to achieve a prescribed backward error. We also compare results on inexact matrix-vector products and preconditioning with the frameworks of Paige and Greenbaum. For block Lanczos, we analyze Householder orthogonalization and singular-value truncation as the block size changes. Under componentwise and normwise error bounds, the computed coefficients satisfy a controlled local recurrence and an exact block Lanczos relation for a nearby symmetric problem in a larger space. A block form of Paige's identity bounds the overlap with a Ritz vector along its residual coordinate direction. An inter-block recurrence describes the evolution of overlap, and a Gram-matrix argument gives a count of additional nearby Ritz values.

Coarse-grained linguistic information, such as named entities or phrases, facilitates adequately representation learning in pre-training. Previous works mainly focus on extending the objective of BERT's Masked Language Modeling (MLM) from masking individual tokens to contiguous sequences of n tokens. We argue that such contiguously masking method neglects to model the intra-dependencies and inter-relation of coarse-grained linguistic information. As an alternative, we propose ERNIE-Gram, an explicitly n-gram masking method to enhance the integration of coarse-grained information into pre-training. In ERNIE-Gram, n-grams are masked and predicted directly using explicit n-gram identities rather than contiguous sequences of n tokens. Furthermore, ERNIE-Gram employs a generator model to sample plausible n-gram identities as optional n-gram masks and predict them in both coarse-grained and fine-grained manners to enable comprehensive n-gram prediction and relation modeling. We pre-train ERNIE-Gram on English and Chinese text corpora and fine-tune on 19 downstream tasks. Experimental results show that ERNIE-Gram outperforms previous pre-training models like XLNet and RoBERTa by a large margin, and achieves comparable results with state-of-the-art methods. The source codes and pre-trained models have been released at https://github.com/PaddlePaddle/ERNIE.

This thesis is divided into three parts. The first part deals with cylindric plane partitions. The second with lambda-determinants and the third with commutators in semi-circular systems. For more detailed abstract please see inside. Cylindric plane partitions may be thought of as a natural generalization of reverse plane partitions. A generating series for the enumeration of cylindric plane partitions was recently given by Borodin. The first result of section one is a new bijective proof of Borodin's identity which makes use of Fomin's growth diagram framework for generalized RSK correspondences. The second result is a (q,t)-analog of Borodin's identity which extends previous work by Okada in the reverse plane partition case. The third result is an explicit combinatorial interpretation of the Macdonald weight occurring in the (q,t)-analog using the non-intersecting lattice path model for cylindric plane partitions. Alternating sign matrices were discovered by Robbins and Rumsey whilst studying λ-determinants. In the second part of this thesis we prove a multi-parameter generalization of the λ-determinant, generalizing a recent result by di Francesco. Like the original λ-determinant, our formula exhibits the Laurent phenomenon. Semicircular systems were first introduced by Voiculescu as a part of his study of von Neumann algebras. In the third part of this thesis we study certain commutator subalgebras of the semicircular system. We find a projection matrix with an interesting self-similar structure. Making use of our projection formula we given an alternative, elementary proof that the semicircular system is a factor.

Vector Symbolic Architectures (VSAs) give a way to represent a complex object as a single fixed-length vector, so that similar objects have similar vector representations. These vector representations then become easy to use for machine learning or nearest-neighbor search. We review a previously proposed VSA method, MBAT (Matrix Binding of Additive Terms), which uses multiplication by random matrices for binding related terms. However, multiplying by such matrices introduces instabilities which can harm performance. Making the random matrices be orthogonal matrices provably fixes this problem. With respect to larger scale applications, we see how to apply MBAT vector representations for any data expressed in JSON. JSON is used in numerous programming languages to express complex data, but its native format appears highly unsuited for machine learning. Expressing JSON as a fixed-length vector makes it readily usable for machine learning and nearest-neighbor search. Creating such JSON vectors also shows that a VSA needs to employ binding operations that are non-commutative. VSAs are now ready to try with full-scale practical applications, including healthcare, pharmaceuticals, and genomics. Keywords: MBAT (Matrix Binding of Additive Terms), VSA (Vector Symbolic Architecture), HDC (Hyperdimensional Computing), Distributed Representations, Binding, Orthogonal Matrices, Recurrent Connections, Machine Learning, Search, JSON, VSA Applications

When dealing with electro or magnetoencephalography records, many supervised prediction tasks are solved by working with covariance matrices to summarize the signals. Learning with these matrices requires using Riemanian geometry to account for their structure. In this paper, we propose a new method to deal with distributions of covariance matrices and demonstrate its computational efficiency on M/EEG multivariate time series. More specifically, we define a Sliced-Wasserstein distance between measures of symmetric positive definite matrices that comes with strong theoretical guarantees. Then, we take advantage of its properties and kernel methods to apply this distance to brain-age prediction from MEG data and compare it to state-of-the-art algorithms based on Riemannian geometry. Finally, we show that it is an efficient surrogate to the Wasserstein distance in domain adaptation for Brain Computer Interface applications.

Large neural networks excel in many domains, but they are expensive to train and fine-tune. A popular approach to reduce their compute or memory requirements is to replace dense weight matrices with structured ones (e.g., sparse, low-rank, Fourier transform). These methods have not seen widespread adoption (1) in end-to-end training due to unfavorable efficiency--quality tradeoffs, and (2) in dense-to-sparse fine-tuning due to lack of tractable algorithms to approximate a given dense weight matrix. To address these issues, we propose a class of matrices (Monarch) that is hardware-efficient (they are parameterized as products of two block-diagonal matrices for better hardware utilization) and expressive (they can represent many commonly used transforms). Surprisingly, the problem of approximating a dense weight matrix with a Monarch matrix, though nonconvex, has an analytical optimal solution. These properties of Monarch matrices unlock new ways to train and fine-tune sparse and dense models. We empirically validate that Monarch can achieve favorable accuracy-efficiency tradeoffs in several end-to-end sparse training applications: speeding up ViT and GPT-2 training on ImageNet classification and Wikitext-103 language modeling by 2x with comparable model quality, and reducing the error on PDE solving and MRI reconstruction tasks by 40%. In sparse-to-dense training, with a simple technique called "reverse sparsification," Monarch matrices serve as a useful intermediate representation to speed up GPT-2 pretraining on OpenWebText by 2x without quality drop. The same technique brings 23% faster BERT pretraining than even the very optimized implementation from Nvidia that set the MLPerf 1.1 record. In dense-to-sparse fine-tuning, as a proof-of-concept, our Monarch approximation algorithm speeds up BERT fine-tuning on GLUE by 1.7x with comparable accuracy.

Given only a few observed entries from a low-rank matrix X, matrix completion is the problem of imputing the missing entries, and it formalizes a wide range of real-world settings that involve estimating missing data. However, when there are too few observed entries to complete the matrix, what other aspects of the underlying matrix can be reliably recovered? We study one such problem setting, that of "one-sided" matrix completion, where our goal is to recover the right singular vectors of X, even in the regime where recovering the left singular vectors is impossible, which arises when there are more rows than columns and very few observations. We propose a natural algorithm that involves imputing the missing values of the matrix X^TX and show that even with only two observations per row in X, we can provably recover X^TX as long as we have at least Omega(r^2 d log d) rows, where r is the rank and d is the number of columns. We evaluate our algorithm on one-sided recovery of synthetic data and low-coverage genome sequencing. In these settings, our algorithm substantially outperforms standard matrix completion and a variety of direct factorization methods.

We provide a full characterisation of all of the possible group equivariant neural networks whose layers are some tensor power of R^{n} for three symmetry groups that are missing from the machine learning literature: O(n), the orthogonal group; SO(n), the special orthogonal group; and Sp(n), the symplectic group. In particular, we find a spanning set of matrices for the learnable, linear, equivariant layer functions between such tensor power spaces in the standard basis of R^{n} when the group is O(n) or SO(n), and in the symplectic basis of R^{n} when the group is Sp(n).

With natural language processing (NLP), researchers aim to enable computers to identify and understand patterns in human languages. This is often difficult because a language embeds many dynamic and varied properties in its syntax, pragmatics and phonology, which need to be captured and processed. The capacity of computers to process natural languages is increasing because NLP researchers are pushing its boundaries. But these research works focus more on well-resourced languages such as English, Japanese, German, French, Russian, Mandarin Chinese, etc. Over 95% of the world's 7000 languages are low-resourced for NLP, i.e. they have little or no data, tools, and techniques for NLP work. In this thesis, we present an overview of diacritic ambiguity and a review of previous diacritic disambiguation approaches on other languages. Focusing on the Igbo language, we report the steps taken to develop a flexible framework for generating datasets for diacritic restoration. Three main approaches, the standard n-gram model, the classification models and the embedding models were proposed. The standard n-gram models use a sequence of previous words to the target stripped word as key predictors of the correct variants. For the classification models, a window of words on both sides of the target stripped word was used. The embedding models compare the similarity scores of the combined context word embeddings and the embeddings of each of the candidate variant vectors.

In the classical transformer attention scheme, we are given three n times d size matrices Q, K, V (the query, key, and value tokens), and the goal is to compute a new n times d size matrix D^{-1} exp(QK^top) V where D = diag( exp(QK^top) {bf 1}_n ). In this work, we study a generalization of attention which captures triple-wise correlations. This generalization is able to solve problems about detecting triple-wise connections that were shown to be impossible for transformers. The potential downside of this generalization is that it appears as though computations are even more difficult, since the straightforward algorithm requires cubic time in n. However, we show that in the bounded-entry setting (which arises in practice, and which is well-studied in both theory and practice), there is actually a near-linear time algorithm. More precisely, we show that bounded entries are both necessary and sufficient for quickly performing generalized computations: bullet On the positive side, if all entries of the input matrices are bounded above by o(sqrt[3]{log n}) then we show how to approximate the ``tensor-type'' attention matrix in n^{1+o(1)} time. bullet On the negative side, we show that if the entries of the input matrices may be as large as Omega(sqrt[3]{log n}), then there is no algorithm that runs faster than n^{3-o(1)} (assuming the Strong Exponential Time Hypothesis from fine-grained complexity theory). We also show that our construction, algorithms, and lower bounds naturally generalize to higher-order tensors and correlations. Interestingly, the higher the order of the tensors, the lower the bound on the entries needs to be for an efficient algorithm. Our results thus yield a natural tradeoff between the boundedness of the entries, and order of the tensor one may use for more expressive, efficient attention computation.

The algebraic diversity framework generalizes temporal averaging over multiple observations to algebraic group action on a single observation for second-order statistical estimation. The central open problem in this framework is group selection: given an M-dimensional observation with unknown covariance structure, find the finite group whose spectral decomposition best matches the covariance. Naive enumeration of all subgroups of the symmetric group S_M requires exponential time in M. We prove that this combinatorial problem reduces to a generalized eigenvalue problem derived from the double commutator of the covariance matrix, yielding a polynomial-time algorithm with complexity O(d^2M^2 + d^3), where d is the dimension of a generator basis. The minimum eigenvector of the double-commutator matrix directly constructs the optimal group generator in closed form, with no iterative optimization. The reduction is exact: the double-commutator minimum eigenvalue is zero if and only if the optimal generator lies in the span of the basis, and its magnitude provides a certifiable optimality gap when it does not. This problem does not appear in the standard catalogs of computational complexity (Garey and Johnson, 1979) and represents a new class linking group theory, matrix analysis, and statistical estimation. We establish connections to independent component analysis (JADE), structured matrix nearness problems, and simultaneous matrix diagonalization, and we show that the double-commutator formulation is the unique approach that is simultaneously polynomial-time, closed-form, and certifiable. We extend the framework to non-Abelian symmetry recovery via a Sequential GEVP with deflation, and add two identifiability theorems characterizing the commutant-lattice ambiguity and the dichotomy on whether Aut(R) recovers a generative subgroup or only a supergroup.

The power of Large Language Models (LLMs) has led us to investigate how they might be fine-tuned for learning the "language of molecular geometry". The fine-tuning of the LLMs using Cartesian coordinates or Z-matrices provides an extremely simple method for accurately predicting equilibrium structures and diverse sets of conformers of small organic and drug-like molecules with excellent accuracy and outperforming specialized deep learning models. While the most common representation of molecular geometries are Cartesian coordinates performs adequately, we find that the inherent invariances and relational nature of geometries represented as Z-matrices provides a better grammar for LLM adaptation. Finally, we show that enhancing an LLMs capabilities for robust prediction of small molecule geometries still retains nearly all of its pre-trained language abilities by randomly mixing in small quantities of natural language prompt-response pairs into the fine-tuning.

Let F be a field, and n geq r>0 be integers, with r even. Denote by A_n(F) the space of all n-by-n alternating matrices with entries in F. We consider the problem of determining the greatest possible dimension for an affine subspace of A_n(F) in which every matrix has rank equal to r (or rank at least r). Recently Rubei has solved this problem over the field of real numbers. We extend her result to all fields with large enough cardinality. Provided that n geq r+3 and |F|geq minbigl(r-1,r{2}+2bigr), we also determine the affine subspaces of rank r matrices in A_n(F) that have the greatest possible dimension, and we point to difficulties for the corresponding problem in the case nleq r+2.

We introduce a simple algorithm that efficiently computes tensor products of Pauli matrices. This is done by tailoring the calculations to this specific case, which allows to avoid unnecessary calculations. The strength of this strategy is benchmarked against state-of-the-art techniques, showing a remarkable acceleration. As a side product, we provide an optimized method for one key calculus in quantum simulations: the Pauli basis decomposition of Hamiltonians.

This paper gives three formulas for the pseudoinverse of a matrix product A = CR. The first is sometimes correct, the second is always correct, and the third is almost never correct. But that third randomized pseudoinverse A^+_r may be very useful when A is a very large matrix. 1. A^+ = R^+C^+ when A = CR and C has independent columns and R has independent rows. 2. A^+ = (C^+CR)^+(CRR^+)^+ is always correct. 3. A^+_r = (P^TCR)^+P^TCRQ(CRQ)^+ = A^+ only when rank(P^TA) = rank(AQ) = rank(A) with A = CR.

We investigate the properties of uniform doubly stochastic random matrices, that is non-negative matrices conditioned to have their rows and columns sum to 1. The rescaled marginal distributions are shown to converge to exponential distributions and indeed even large sub-matrices of side-length o(n^{1/2-ε}) behave like independent exponentials. We determine the limiting empirical distribution of the singular values the the matrix. Finally the mixing time of the associated Markov chains is shown to be exactly 2 with high probability.

The GLEU metric was proposed for evaluating grammatical error corrections using n-gram overlap with a set of reference sentences, as opposed to precision/recall of specific annotated errors (Napoles et al., 2015). This paper describes improvements made to the GLEU metric that address problems that arise when using an increasing number of reference sets. Unlike the originally presented metric, the modified metric does not require tuning. We recommend that this version be used instead of the original version.

Drug target interaction (DTI) prediction is a cornerstone of computational drug discovery, enabling rational design, repurposing, and mechanistic insights. While deep learning has advanced DTI modeling, existing approaches primarily rely on SMILES protein pairs and fail to exploit the rich multimodal information available for small molecules and proteins. We introduce GRAMDTI, a pretraining framework that integrates multimodal molecular and protein inputs into unified representations. GRAMDTI extends volume based contrastive learning to four modalities, capturing higher-order semantic alignment beyond conventional pairwise approaches. To handle modality informativeness, we propose adaptive modality dropout, dynamically regulating each modality's contribution during pre-training. Additionally, IC50 activity measurements, when available, are incorporated as weak supervision to ground representations in biologically meaningful interaction strengths. Experiments on four publicly available datasets demonstrate that GRAMDTI consistently outperforms state of the art baselines. Our results highlight the benefits of higher order multimodal alignment, adaptive modality utilization, and auxiliary supervision for robust and generalizable DTI prediction.

Recommendation is the task of ranking items (e.g. movies or products) according to individual user needs. Current systems rely on collaborative filtering and content-based techniques, which both require structured training data. We propose a framework for recommendation with off-the-shelf pretrained language models (LM) that only used unstructured text corpora as training data. If a user u liked Matrix and Inception, we construct a textual prompt, e.g. "Movies like Matrix, Inception, {<m{>}"} to estimate the affinity between u and m with LM likelihood. We motivate our idea with a corpus analysis, evaluate several prompt structures, and we compare LM-based recommendation with standard matrix factorization trained on different data regimes. The code for our experiments is publicly available (https://colab.research.google.com/drive/1f1mlZ-FGaLGdo5rPzxf3vemKllbh2esT?usp=sharing).

We give a classical analogue to Kerenidis and Prakash's quantum recommendation system, previously believed to be one of the strongest candidates for provably exponential speedups in quantum machine learning. Our main result is an algorithm that, given an m times n matrix in a data structure supporting certain ell^2-norm sampling operations, outputs an ell^2-norm sample from a rank-k approximation of that matrix in time O(poly(k)log(mn)), only polynomially slower than the quantum algorithm. As a consequence, Kerenidis and Prakash's algorithm does not in fact give an exponential speedup over classical algorithms. Further, under strong input assumptions, the classical recommendation system resulting from our algorithm produces recommendations exponentially faster than previous classical systems, which run in time linear in m and n. The main insight of this work is the use of simple routines to manipulate ell^2-norm sampling distributions, which play the role of quantum superpositions in the classical setting. This correspondence indicates a potentially fruitful framework for formally comparing quantum machine learning algorithms to classical machine learning algorithms.

When rows of an n times d matrix A are given in a stream, we study algorithms for approximating the top eigenvector of the matrix {A}^TA (equivalently, the top right singular vector of A). We consider worst case inputs A but assume that the rows are presented to the streaming algorithm in a uniformly random order. We show that when the gap parameter R = σ_1(A)^2/σ_2(A)^2 = Ω(1), then there is a randomized algorithm that uses O(h cdot d cdot polylog(d)) bits of space and outputs a unit vector v that has a correlation 1 - O(1/R) with the top eigenvector v_1. Here h denotes the number of heavy rows in the matrix, defined as the rows with Euclidean norm at least |{A}|_F/d cdot operatorname{polylog(d)}. We also provide a lower bound showing that any algorithm using O(hd/R) bits of space can obtain at most 1 - Ω(1/R^2) correlation with the top eigenvector. Thus, parameterizing the space complexity in terms of the number of heavy rows is necessary for high accuracy solutions. Our results improve upon the R = Ω(log n cdot log d) requirement in a recent work of Price and Xun (FOCS 2024). We note that the algorithm of Price and Xun works for arbitrary order streams whereas our algorithm requires a stronger assumption that the rows are presented in a uniformly random order. We additionally show that the gap requirements in their analysis can be brought down to R = Ω(log^2 d) for arbitrary order streams and R = Ω(log d) for random order streams. The requirement of R = Ω(log d) for random order streams is nearly tight for their analysis as we obtain a simple instance with R = Ω(log d/loglog d) for which their algorithm, with any fixed learning rate, cannot output a vector approximating the top eigenvector v_1.

We present a statistical model for German medical natural language processing trained for named entity recognition (NER) as an open, publicly available model. The work serves as a refined successor to our first GERNERMED model which is substantially outperformed by our work. We demonstrate the effectiveness of combining multiple techniques in order to achieve strong results in entity recognition performance by the means of transfer-learning on pretrained deep language models (LM), word-alignment and neural machine translation. Due to the sparse situation on open, public medical entity recognition models for German texts, this work offers benefits to the German research community on medical NLP as a baseline model. Since our model is based on public English data, its weights are provided without legal restrictions on usage and distribution. The sample code and the statistical model is available at: https://github.com/frankkramer-lab/GERNERMED-pp

Spectral methods are popular in detecting global structures in the given data that can be represented as a matrix. However when the data matrix is sparse or noisy, classic spectral methods usually fail to work, due to localization of eigenvectors (or singular vectors) induced by the sparsity or noise. In this work, we propose a general method to solve the localization problem by learning a regularization matrix from the localized eigenvectors. Using matrix perturbation analysis, we demonstrate that the learned regularizations suppress down the eigenvalues associated with localized eigenvectors and enable us to recover the informative eigenvectors representing the global structure. We show applications of our method in several inference problems: community detection in networks, clustering from pairwise similarities, rank estimation and matrix completion problems. Using extensive experiments, we illustrate that our method solves the localization problem and works down to the theoretical detectability limits in different kinds of synthetic data. This is in contrast with existing spectral algorithms based on data matrix, non-backtracking matrix, Laplacians and those with rank-one regularizations, which perform poorly in the sparse case with noise.

Differential geometric approaches to the analysis and processing of data in the form of symmetric positive definite (SPD) matrices have had notable successful applications to numerous fields including computer vision, medical imaging, and machine learning. The dominant geometric paradigm for such applications has consisted of a few Riemannian geometries associated with spectral computations that are costly at high scale and in high dimensions. We present a route to a scalable geometric framework for the analysis and processing of SPD-valued data based on the efficient computation of extreme generalized eigenvalues through the Hilbert and Thompson geometries of the semidefinite cone. We explore a particular geodesic space structure based on Thompson geometry in detail and establish several properties associated with this structure. Furthermore, we define a novel iterative mean of SPD matrices based on this geometry and prove its existence and uniqueness for a given finite collection of points. Finally, we state and prove a number of desirable properties that are satisfied by this mean.

Data-driven techniques have a large potential to transform and accelerate the chemical sciences. However, chemical sciences also pose the unique challenge of very diverse, small, fuzzy datasets that are difficult to leverage in conventional machine learning approaches completely. A new class of models, general-purpose models (GPMs) such as large language models, have shown the ability to solve tasks they have not been directly trained on, and to flexibly operate with low amounts of data in different formats. In this review, we discuss fundamental building principles of GPMs and review recent applications of those models in the chemical sciences across the entire scientific process. While many of these applications are still in the prototype phase, we expect that the increasing interest in GPMs will make many of them mature in the coming years.

Lookup-based memory has been a promising way to scale the parameters of large language models (LLMs). It retrieves learned representations of local token patterns, such as n-grams, instead of reconstructing them through successive layers of computation. However, existing designs such as Engram treat each retrieved embedding as a monolithic unit. Each embedding is stored in its own hashed slot and modulated by a single scalar gate. As a result, polysemous patterns cannot selectively read out the components of their memory that are relevant to the context. Moreover, parameters are shared only through hash collisions, which are largely unrelated to semantics. We propose FactorEngram, a factorized n-gram memory with basis-level contextual gating. FactorEngram retrieves sparsity-regularized coefficients over a dictionary of basis vectors shared across patterns, so related patterns can reuse common components. The same dictionary is also used for gating. The backbone hidden state is scored against each basis vector to gate the corresponding coefficient before reconstruction, which lets the context modulate each memory component individually. FactorEngram also covers both individual tokens and multi-token n-grams, and we systematically study where the memory branch should be inserted. On 340M- and 1B-parameter Transformer backbones, FactorEngram improves language modeling and downstream task performance. Ablation studies confirm the contribution of each component and identify insertion before the attention sublayer in the middle layers as an effective configuration.

This book introduces the mathematical foundations and techniques that lead to the development and analysis of many of the algorithms that are used in machine learning. It starts with an introductory chapter that describes notation used throughout the book and serve at a reminder of basic concepts in calculus, linear algebra and probability and also introduces some measure theoretic terminology, which can be used as a reading guide for the sections that use these tools. The introductory chapters also provide background material on matrix analysis and optimization. The latter chapter provides theoretical support to many algorithms that are used in the book, including stochastic gradient descent, proximal methods, etc. After discussing basic concepts for statistical prediction, the book includes an introduction to reproducing kernel theory and Hilbert space techniques, which are used in many places, before addressing the description of various algorithms for supervised statistical learning, including linear methods, support vector machines, decision trees, boosting, or neural networks. The subject then switches to generative methods, starting with a chapter that presents sampling methods and an introduction to the theory of Markov chains. The following chapter describe the theory of graphical models, an introduction to variational methods for models with latent variables, and to deep-learning based generative models. The next chapters focus on unsupervised learning methods, for clustering, factor analysis and manifold learning. The final chapter of the book is theory-oriented and discusses concentration inequalities and generalization bounds.

In this study, we propose a simple method for fault-tolerant Strassen-like matrix multiplications. The proposed method is based on using two distinct Strassen-like algorithms instead of replicating a given one. We have realized that using two different algorithms, new check relations arise resulting in more local computations. These local computations are found using computer aided search. To improve performance, special parity (extra) sub-matrix multiplications (PSMMs) are generated (two of them) at the expense of increasing communication/computation cost of the system. Our preliminary results demonstrate that the proposed method outperforms a Strassen-like algorithm with two copies and secures a very close performance to three copy version using only 2 PSMMs, reducing the total number of compute nodes by around 24\% i.e., from 21 to 16.

Recently, deep clustering methods have gained momentum because of the high representational power of deep neural networks (DNNs) such as autoencoder. The key idea is that representation learning and clustering can reinforce each other: Good representations lead to good clustering while good clustering provides good supervisory signals to representation learning. Critical questions include: 1) How to optimize representation learning and clustering? 2) Should the reconstruction loss of autoencoder be considered always? In this paper, we propose DEKM (for Deep Embedded K-Means) to answer these two questions. Since the embedding space generated by autoencoder may have no obvious cluster structures, we propose to further transform the embedding space to a new space that reveals the cluster-structure information. This is achieved by an orthonormal transformation matrix, which contains the eigenvectors of the within-class scatter matrix of K-means. The eigenvalues indicate the importance of the eigenvectors' contributions to the cluster-structure information in the new space. Our goal is to increase the cluster-structure information. To this end, we discard the decoder and propose a greedy method to optimize the representation. Representation learning and clustering are alternately optimized by DEKM. Experimental results on the real-world datasets demonstrate that DEKM achieves state-of-the-art performance.

Writing systems of Indic languages have orthographic syllables, also known as complex graphemes, as unique horizontal units. A prominent feature of these languages is these complex grapheme units that comprise consonants/consonant conjuncts, vowel diacritics, and consonant diacritics, which, together make a unique Language. Unicode-based writing schemes of these languages often disregard this feature of these languages and encode words as linear sequences of Unicode characters using an intricate scheme of connector characters and font interpreters. Due to this way of using a few dozen Unicode glyphs to write thousands of different unique glyphs (complex graphemes), there are serious ambiguities that lead to malformed words. In this paper, we are proposing two libraries: i) a normalizer for normalizing inconsistencies caused by a Unicode-based encoding scheme for Indic languages and ii) a grapheme parser for Abugida text. It deconstructs words into visually distinct orthographic syllables or complex graphemes and their constituents. Our proposed normalizer is a more efficient and effective tool than the previously used IndicNLP normalizer. Moreover, our parser and normalizer are also suitable tools for general Abugida text processing as they performed well in our robust word-based and NLP experiments. We report the pipeline for the scripts of 7 languages in this work and develop the framework for the integration of more scripts.

This paper presents a computationally efficient machine-learned method for natural language response suggestion. Feed-forward neural networks using n-gram embedding features encode messages into vectors which are optimized to give message-response pairs a high dot-product value. An optimized search finds response suggestions. The method is evaluated in a large-scale commercial e-mail application, Inbox by Gmail. Compared to a sequence-to-sequence approach, the new system achieves the same quality at a small fraction of the computational requirements and latency.

Grammatical error correction (GEC) is the task of correcting typos, spelling, punctuation and grammatical issues in text. Approaching the problem as a sequence-to-sequence task, we compare the use of a common subword unit vocabulary and byte-level encoding. Initial synthetic training data is created using an error-generating pipeline, and used for finetuning two subword-level models and one byte-level model. Models are then finetuned further on hand-corrected error corpora, including texts written by children, university students, dyslexic and second-language writers, and evaluated over different error types and origins. We show that a byte-level model enables higher correction quality than a subword approach, not only for simple spelling errors, but also for more complex semantic, stylistic and grammatical issues. In particular, initial training on synthetic corpora followed by finetuning on a relatively small parallel corpus of real-world errors helps the byte-level model correct a wide range of commonly occurring errors. Our experiments are run for the Icelandic language but should hold for other similar languages, particularly morphologically rich ones.

With the increasing number of mobile devices, there has been continuous research on generating optimized Language Models (LMs) for soft keyboard. In spite of advances in this domain, building a single LM for low-end feature phones as well as high-end smartphones is still a pressing need. Hence, we propose a novel technique, Optimized N-gram (Op-Ngram), an end-to-end N-gram pipeline that utilises mobile resources efficiently for faster Word Completion (WC) and Next Word Prediction (NWP). Op-Ngram applies Stupid Backoff and pruning strategies to generate a light-weight model. The LM loading time on mobile is linear with respect to model size. We observed that Op-Ngram gives 37% improvement in Language Model (LM)-ROM size, 76% in LM-RAM size, 88% in loading time and 89% in average suggestion time as compared to SORTED array variant of BerkeleyLM. Moreover, our method shows significant performance improvement over KenLM as well.

Modelling compositionality has been a longstanding area of research in the field of vector space semantics. The categorical approach to compositionality maps grammar onto vector spaces in a principled way, but comes under fire for requiring the formation of very high-dimensional matrices and tensors, and therefore being computationally infeasible. In this paper I show how a linear simplification of recursive neural tensor network models can be mapped directly onto the categorical approach, giving a way of computing the required matrices and tensors. This mapping suggests a number of lines of research for both categorical compositional vector space models of meaning and for recursive neural network models of compositionality.

Generative Adversarial Networks (GANs), especially the recent style-based generators (StyleGANs), have versatile semantics in the structured latent space. Latent semantics discovery methods emerge to move around the latent code such that only one factor varies during the traversal. Recently, an unsupervised method proposed a promising direction to directly use the eigenvectors of the projection matrix that maps latent codes to features as the interpretable directions. However, one overlooked fact is that the projection matrix is non-orthogonal and the number of eigenvectors is too large. The non-orthogonality would entangle semantic attributes in the top few eigenvectors, and the large dimensionality might result in meaningless variations among the directions even if the matrix is orthogonal. To avoid these issues, we propose Householder Projector, a flexible and general low-rank orthogonal matrix representation based on Householder transformations, to parameterize the projection matrix. The orthogonality guarantees that the eigenvectors correspond to disentangled interpretable semantics, while the low-rank property encourages that each identified direction has meaningful variations. We integrate our projector into pre-trained StyleGAN2/StyleGAN3 and evaluate the models on several benchmarks. Within only 1% of the original training steps for fine-tuning, our projector helps StyleGANs to discover more disentangled and precise semantic attributes without sacrificing image fidelity.

Objective: Electroencephalography signals are recorded as a multidimensional dataset. We propose a new framework based on the augmented covariance extracted from an autoregressive model to improve motor imagery classification. Methods: From the autoregressive model can be derived the Yule-Walker equations, which show the emergence of a symmetric positive definite matrix: the augmented covariance matrix. The state-of the art for classifying covariance matrices is based on Riemannian Geometry. A fairly natural idea is therefore to extend the standard approach using these augmented covariance matrices. The methodology for creating the augmented covariance matrix shows a natural connection with the delay embedding theorem proposed by Takens for dynamical systems. Such an embedding method is based on the knowledge of two parameters: the delay and the embedding dimension, respectively related to the lag and the order of the autoregressive model. This approach provides new methods to compute the hyper-parameters in addition to standard grid search. Results: The augmented covariance matrix performed noticeably better than any state-of-the-art methods. We will test our approach on several datasets and several subjects using the MOABB framework, using both within-session and cross-session evaluation. Conclusion: The improvement in results is due to the fact that the augmented covariance matrix incorporates not only spatial but also temporal information, incorporating nonlinear components of the signal through an embedding procedure, which allows the leveraging of dynamical systems algorithms. Significance: These results extend the concepts and the results of the Riemannian distance based classification algorithm.

Transformers can learn to perform numerical computations from examples only. I study nine problems of linear algebra, from basic matrix operations to eigenvalue decomposition and inversion, and introduce and discuss four encoding schemes to represent real numbers. On all problems, transformers trained on sets of random matrices achieve high accuracies (over 90%). The models are robust to noise, and can generalize out of their training distribution. In particular, models trained to predict Laplace-distributed eigenvalues generalize to different classes of matrices: Wigner matrices or matrices with positive eigenvalues. The reverse is not true.

In recent years, spectral clustering has become one of the most popular modern clustering algorithms. It is simple to implement, can be solved efficiently by standard linear algebra software, and very often outperforms traditional clustering algorithms such as the k-means algorithm. On the first glance spectral clustering appears slightly mysterious, and it is not obvious to see why it works at all and what it really does. The goal of this tutorial is to give some intuition on those questions. We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. Advantages and disadvantages of the different spectral clustering algorithms are discussed.

We investigate LoRA in federated learning through the lens of the asymmetry analysis of the learned A and B matrices. In doing so, we uncover that A matrices are responsible for learning general knowledge, while B matrices focus on capturing client-specific knowledge. Based on this finding, we introduce Federated Share-A Low-Rank Adaptation (FedSA-LoRA), which employs two low-rank trainable matrices A and B to model the weight update, but only A matrices are shared with the server for aggregation. Moreover, we delve into the relationship between the learned A and B matrices in other LoRA variants, such as rsLoRA and VeRA, revealing a consistent pattern. Consequently, we extend our FedSA-LoRA method to these LoRA variants, resulting in FedSA-rsLoRA and FedSA-VeRA. In this way, we establish a general paradigm for integrating LoRA with FL, offering guidance for future work on subsequent LoRA variants combined with FL. Extensive experimental results on natural language understanding and generation tasks demonstrate the effectiveness of the proposed method.

Proposing an effective and flexible matrix to represent a graph is a fundamental challenge that has been explored from multiple perspectives, e.g., filtering in Graph Fourier Transforms. In this work, we develop a novel and general framework which unifies many existing GNN models from the view of parameterized decomposition and filtering, and show how it helps to enhance the flexibility of GNNs while alleviating the smoothness and amplification issues of existing models. Essentially, we show that the extensively studied spectral graph convolutions with learnable polynomial filters are constrained variants of this formulation, and releasing these constraints enables our model to express the desired decomposition and filtering simultaneously. Based on this generalized framework, we develop models that are simple in implementation but achieve significant improvements and computational efficiency on a variety of graph learning tasks. Code is available at https://github.com/qslim/PDF.

The reciprocal distance Laplacian matrix of a connected graph G is defined as RD^L(G)=RT(G)-RD(G), where RT(G) is the diagonal matrix of reciprocal distance degrees and RD(G) is the Harary matrix. Since RD^L(G) is a real symmetric matrix, we denote its eigenvalues as λ_1(RD^L(G))geq λ_2(RD^L(G))geq dots geq λ_n(RD^L(G)). The largest eigenvalue λ_1(RD^L(G)) of RD^L(G) is called the reciprocal distance Laplacian spectral radius. In this article, we prove that the multiplicity of n as a reciprocal distance Laplacian eigenvalue of RD^L(G) is exactly one less than the number of components in the complement graph G of G. We show that the class of the complete bipartite graphs maximize the reciprocal distance Laplacian spectral radius among all the bipartite graphs with n vertices. Also, we show that the star graph S_n is the unique graph having the maximum reciprocal distance Laplacian spectral radius in the class of trees with n vertices. We determine the reciprocal distance Laplacian spectrum of several well known graphs. We prove that the complete graph K_n, K_n-e, the star S_n, the complete balanced bipartite graph K_{n{2},n{2}} and the complete split graph CS(n,α) are all determined from the RD^L-spectrum.

When you merge two fine-tuned models from the same base checkpoint by simply averaging their weights, you implicitly assume their attention subspaces are still aligned. Recent work attempts to fix misalignments by learning an invertible correction matrix, M, for each model's query and key projections. This correction cancels out---using M on the query side and M^{-T} on the key side---right before the dot product. However, this cancellation is only exact if nothing sits between the projection and the dot product. In reality, almost all modern open-weight language models put a rotary position embedding (RoPE) exactly there. In this paper, we show that this cancellation is exact under RoPE if and only if M commutes with RoPE's per-position rotation. We derive the specific class of matrices where this holds: a scaled rotation acting independently within each RoPE frequency pair. This forms a strict, low-dimensional subset of the unconstrained matrices that current methods normally train. Building on this, we turn this constrained matrix class into a new merging method. While keeping the base weights entirely frozen, two fine-tunes each learn their own RoPE-compliant correction matrices. We optimize these corrections against a chosen blend ratio so the final result can be adjusted post-hoc like a dial, rather than locked into a single fixed merge. Our default approach trains at one fixed blend ratio, similar to how LoRA sets its scaling hyperparameter in advance. We also experiment with resampling the blend ratio randomly at every training step, and we report the results of both approaches.

We show that large language models (LLMs) can be used to distinguish the writings of different authors. Specifically, an individual GPT-2 model, trained from scratch on the works of one author, will predict held-out text from that author more accurately than held-out text from other authors. We suggest that, in this way, a model trained on one author's works embodies the unique writing style of that author. We first demonstrate our approach on books written by eight different (known) authors. We also use this approach to confirm R. P. Thompson's authorship of the well-studied 15th book of the Oz series, originally attributed to F. L. Baum.

We study the estimation of a planted signal hidden in a recently introduced nested matrix-tensor model, which is an extension of the classical spiked rank-one tensor model, motivated by multi-view clustering. Prior work has theoretically examined the performance of a tensor-based approach, which relies on finding a best rank-one approximation, a problem known to be computationally hard. A tractable alternative approach consists in computing instead the best rank-one (matrix) approximation of an unfolding of the observed tensor data, but its performance was hitherto unknown. We quantify here the performance gap between these two approaches, in particular by deriving the precise algorithmic threshold of the unfolding approach and demonstrating that it exhibits a BBP-type transition behavior. This work is therefore in line with recent contributions which deepen our understanding of why tensor-based methods surpass matrix-based methods in handling structured tensor data.

Large collections of matrices arise throughout modern machine learning, signal processing, and scientific computing, where they are commonly compressed by concatenation followed by truncated singular value decomposition (SVD). This strategy enables parameter sharing and efficient reconstruction and has been widely adopted across domains ranging from multi-view learning and signal processing to neural network compression. However, it leaves a fundamental question unanswered: which matrices can be safely concatenated and compressed together under explicit reconstruction error constraints? Existing approaches rely on heuristic or architecture-specific grouping and provide no principled guarantees on the resulting SVD approximation error. In the present work, we introduce a theory-driven framework for compression-aware clustering of matrices under SVD compression constraints. Our analysis establishes new spectral bounds for horizontally concatenated matrices, deriving global upper bounds on the optimal rank-r SVD reconstruction error from lower bounds on singular value growth. The first bound follows from Weyl-type monotonicity under blockwise extensions, while the second leverages singular values of incremental residuals to yield tighter, per-block guarantees. We further develop an efficient approximate estimator based on incremental truncated SVD that tracks dominant singular values without forming the full concatenated matrix. Therefore, we propose three clustering algorithms that merge matrices only when their predicted joint SVD compression error remains below a user-specified threshold. The algorithms span a trade-off between speed, provable accuracy, and scalability, enabling compression-aware clustering with explicit error control. Code is available online.

Sparse matrices, more specifically SpGEMM kernels, are commonly found in a wide range of applications, spanning graph-based path-finding to machine learning algorithms (e.g., neural networks). A particular challenge in implementing SpGEMM kernels has been the pressure placed on DRAM memory. One approach to tackle this problem is to use an inner product method for the SpGEMM kernel implementation. While the inner product produces fewer intermediate results, it can end up saturating the memory bandwidth, given the high number of redundant fetches of the input matrix elements. Using an outer product-based SpGEMM kernel can reduce redundant fetches, but at the cost of increased overhead due to extra computation and memory accesses for producing/managing partial products. In this thesis, we introduce a novel SpGEMM kernel implementation based on the row-wise product approach. We leverage atomic instructions to merge intermediate partial products as they are generated. The use of atomic instructions eliminates the need to create partial product matrices. To evaluate our row-wise product approach, we map an optimized SpGEMM kernel to a custom accelerator designed to accelerate graph-based applications. The targeted accelerator is an experimental system named PIUMA, being developed by Intel. PIUMA provides several attractive features, including fast context switching, user-configurable caches, globally addressable memory, non-coherent caches, and asynchronous pipelines. We tailor our SpGEMM kernel to exploit many of the features of the PIUMA fabric. This thesis compares our SpGEMM implementation against prior solutions, all mapped to the PIUMA framework. We briefly describe some of the PIUMA architecture features and then delve into the details of our optimized SpGEMM kernel. Our SpGEMM kernel can achieve 9.4x speedup as compared to competing approaches.

Why does the low dimensionality of representations, typically dapprox 1000, not prevent modern embedding-based retrieval models from scaling to billions, or even trillions, of data points? To answer this question, we study maximal-margin embeddings in the following retrieval model, classically studied in communication complexity [PS86] and more recently in embedding-based retrieval [WBNL26]. Let Ain {0,1}^{Ntimes n} be a matrix indicating whether each of N queries is relevant to each of n documents. We are interested in the largest margin m>0, denoted by m^{rd}(d, A), for which there exist unit norm embeddings of the queries and documents {U_j}_{j = 1}^N, {V_i}_{i = 1}^n with the following property. langle U_j, V_irangle ge m whenever A_{ji} = 1 and langle U_j, V_irangle le -m otherwise. A large margin is a key proxy for representation quality: it controls both robustness to perturbations and compositional generalization across queries. Our main theorem establishes that the best possible margin without a restriction on the dimension, m^{rd}(+infty, A), can be nearly achieved in dimension d = O(m^{rd}(+infty, A)^{-2}log n) which improves a theorem of [BDES02]. Together with a matching lower bound in Theorem 1.5, we conclude that when Ain {0,1}^{n{k}times n} is the matrix containing all possible k-sparse rows once, dimension d = O(klog (n/k)) is necessary and sufficient for the maximal possible margin m^{rd}(+infty, A) = Θ(k^{-1/2}) in this setting. This fully resolves the setup of [WBNL26]. We also give several constructions for large margins when d = o(klog (n/k)). Finally, we empirically test the InfoNCE and sigmoid losses for producing large margin embeddings and demonstrate a clear advantage of the sigmoid loss.

General Matrix Multiplication (GEMM) is a critical kernel in high-performance computing and deep learning. While modern architectures like ARM's Scalable Matrix Extension (SME) introduce dedicated hardware for matrix operations, existing linear algebra libraries fail to fully exploit its potential, particularly for large matrices. This paper presents MpGEMM, an open-source library that leverages key architectural features of SME to optimize GEMM across multiple precisions. Through a systematic characterization of SME, we derive optimization guidelines that inform our design. MpGEMM employs cache-aware partitioning, efficient data packing with on-the-fly transposition, and specialized micro-kernels that utilize multi-vector loads and all available tile registers. Evaluated on an Apple M4 Pro with real-world workloads from DeepSeek and LLaMA, MpGEMM achieves an average speedup of 1.23x over the vendor-optimized Apple Accelerate library and significantly outperforms other open-source alternatives.

We present BioMatrix, the first multimodal foundation model that natively integrates sequences, structures, and natural language for both molecules and proteins within a single decoder-only architecture. Existing biological foundation models pursue native multimodality and broad entity coverage separately: those that fuse multiple modalities under a shared objective remain confined to a single entity type, while those spanning multiple entity types either omit explicit structural modeling or rely on adapter-based designs in which the model cannot natively generate the very modalities it can read. BioMatrix closes this gap by mapping molecular sequences (supporting both SMILES and SELFIES notations), molecular structures, protein sequences, protein structures, and natural language into a shared discrete token space through a unified tokenization scheme, so that all modalities are consumed and produced uniformly under a single next-token prediction objective -- without external encoders, projection adapters, or modality-specific output heads. Built upon the Qwen3 language model (1.7B and 4B), BioMatrix is continually pretrained on 304.4 billion tokens spanning general and domain-specific text, sequence and structure views of molecules and proteins, and cross-modal corpora that interleave biomolecular entities with scientific text and link distinct entities through molecule-protein and protein-protein interaction data. After tuning on a comprehensive suite of downstream applications covering 80 tasks across 6 categories -- encompassing single-entity and multi-entity understanding and generation tasks across and within modalities -- BioMatrix achieves state-of-the-art or competitive performance on 77 out of 80 tasks, demonstrating that a single, natively multimodal generalist model can effectively match or surpass specialized approaches across a wide range of biological tasks.

We present a new method for constructing Hadamard matrices that combines transformer neural networks with local search in the PatternBoost framework. Our approach is designed for extremely sparse combinatorial search problems and is particularly effective for Hadamard matrices of Goethals--Seidel type, where Fourier methods permit fast scoring and optimisation. For orders between 100 and 200, it produces large numbers of inequivalent Hadamard matrices, and for larger orders, it succeeds where local search from random initialisation fails. The largest example found by our method has order 252. In addition to these new constructions, our experiments reveal that the transformer can discover and exploit useful hidden symmetry in the search space.

Semidefinite programming is a fundamental problem class in convex optimization, but despite recent advances in solvers, solving large-scale semidefinite programs remains challenging. Generally the matrix functions involved are spectral or unitarily invariant, i.e., they depend only on the eigenvalues or singular values of the matrix. This paper investigates how spectral matrix cones -- cones defined from epigraphs and perspectives of spectral or unitarily invariant functions -- can be used to enhance first-order conic solvers for semidefinite programs. Our main result shows that projecting a matrix can be reduced to projecting its eigenvalues or singular values, which we demonstrate can be done at a negligible cost compared to the eigenvalue or singular value decomposition itself. We have integrated support for spectral matrix cone projections into the Splitting Conic Solver (SCS). Numerical experiments show that SCS with this enhancement can achieve speedups of up to an order of magnitude for solving semidefinite programs arising in experimental design, robust principal component analysis, and graph partitioning.