arXiv subjects

Anastasios Zouzias

Publications and source records attributed to Anastasios Zouzias.

At least 19 recordsLinked to original sources

arXiv2026

Approaching I/O-optimality for Approximate Attention

We revisit the I/O complexity of attention in large language models. Given query-key-value matrices $Q,K,V\in\mathbb{R}^{n\times d}$, and a machine with fast memory size $M$, the goal is to compute the "attention matrix" $A=\text{softmax}(Q K ^{\top}/\sqrt{d}) V$ with the minimal number of data transfers between fast and slow memory. Existing methods in the literature, most notably FlashAttention and its variants, incur an I/O cost that depends quadratically on $n$, while a trivial lower bound only requires $\Omega(nd)$ I/O's to read the inputs and write the output. In this work, we present a technique for computing attention where the I/O cost only depends almost-linearly on $n$ in most parameter regimes. This is achieved by developing I/O-efficient algorithms inspired by the recent approximate attention framework of Alman and Song. We also prove corresponding lower bounds in each parameter regime to show that our algorithms are indeed close to I/O-optimal.

cs.LG↗

arXiv2026

Fast and Stable Triangular Inversion for Delta-Rule Linear Transformers

Linear attention has emerged as a cornerstone for efficient long-context architectures, as evidenced by its integration into state-of-the-art open-source models including Qwen3.5/3.6, Kimi Linear, and RWKV-7. Models that incorporate linear attention layers with the so-called Delta-Rule involve the inversion of triangular matrices as a core sub-routine. This operation often forms a performance bottleneck, and, due to its high-sensitivity to numerical errors, it can significantly deteriorate end-to-end model accuracy if it is not carefully implemented. This work provides a systematic analysis of both direct and iterative triangular inversion algorithms, targeting methods that are rich in matrix products, and, therefore, have the potential to efficiently utilize modern hardware. To that end, our analysis covers a broad spectrum of mathematical and practical aspects, with a heavy focus on numerical stability, computational complexity, and, ultimately, hardware efficiency and practical considerations. We provide a rigorous experimental evaluation to verify these properties in practical scenarios, and in low-precision floating-point representations, highlighting the strengths and limitations of each method. Performance benchmarks on NPUs reveal up to $4.3\times$ speed-up against the state-of-the-art implementations of SGLang for triangular matrix inversion, leading to significant performance improvements on the entire layer level, while maintaining full end-to-end model accuracy.

cs.LG↗

arXiv2025

Prefix Sums via Kronecker Products

In this work, we revisit prefix sums through the lens of linear algebra. We describe an identity that decomposes triangular all-ones matrices as a sum of two Kronecker products, and apply it to design recursive prefix sum algorithms and circuits. Notably, the proposed family of circuits is the first one that achieves the following three properties simultaneously: (i) zero-deficiency, (ii) constant fan-out per-level, and (iii) depth that is asymptotically strictly smaller than $2\log(n)$ for input length $n$. As an application, we show how to use these circuits to design quantum adders with $1.893\log(n)+O(1)$ Toffoli depth, $O(n)$ Toffoli gates, and $O(n)$ additional qubits, improving the Toffoli depth and/or Toffoli size of existing constructions.

quant-ph↗

arXiv2025

Segmented Operations using Matrix Multiplications

Specialized computational units that perform small matrix multiplications as primitive operations are typically present in modern AI accelerators. However, these Matrix Multiplication Units (MMUs) are often underutilized for many fundamental deep learning operations besides dense matrix multiplications. Coincidentally, the lack of a rigorous theoretical model of computation for such architectures obstructs algorithmic design. In this work, we propose MMV-RAM, a computational model which judiciously extends the Vector-RAM model with an additional MMU. We provide a detailed theoretical analysis and carefully balance the computational power between the matrix and vector units, guided by the circuit complexity lower bound that parity is not in AC{[0]}. Given MMV-RAM, we proceed to algorithm design, starting with two fundamental parallel operations: segmented scan and sum. By expressing them as compositions of elementary parallel primitives (e.g., seg. sum reduces to: scan, compress, and vector differentiation), we can exploit MMUs to perform speculative blocked computations, ultimately leading to provable theoretical speed-ups against vector-only approaches. These results extend to other ubiquitous AI kernels, including dense matrix product, and sparse matrix-vector product. As a case study, we implemented the proposed algorithms on the Ascend 910B AI accelerator, which contains matrix and vector cores. We evaluate these implementations on synthetic and real-world datasets from various applications, including Large Language Models.

cs.DS↗

arXiv2025

Parallel Scan on Ascend AI Accelerators

We design and implement parallel prefix sum (scan) algorithms using Ascend AI accelerators. Ascend accelerators feature specialized computing units: the cube units for efficient matrix multiplication and the vector units for optimized vector operations. A key feature of the proposed scan algorithms is their extensive use of matrix multiplications and accumulations enabled by the cube unit. To showcase the effectiveness of these algorithms, we also implement and evaluate several scan-based operators commonly used in AI workloads, including sorting, tensor masking, and top-$k$ / top-$p$ sampling. Our single-core results demonstrate substantial performance improvements, with speedups ranging from $5\times$ to $9.6\times$ compared to vector-only implementations for sufficiently large input lengths. Additionally, we present a multi-core scan algorithm that fully utilizes both the cube and vector units of Ascend, reaching up to 74.9\% of the memory bandwidth achieved by memory copy. Furthermore, our radix sort implementation, which utilizes matrix multiplications for its parallel splits, showcases the potential of matrix engines to enhance complex operations, offering up to $3.3\times$ speedup over the vector-only baseline.

cs.DC↗

arXiv2024

A Parallel Scan Algorithm in the Tensor Core Unit Model

We present a parallel scan (prefix sum) algorithm in the Tensor Core Unit (TCU) model of computation. The TCU model assumes that multiplication between two square matrices of constant size $s$ is a basic operation. In the $(s^2, \ell)$-TCU model, we show that for inputs of size $n$, the algorithm has depth at most $2\lfloor \log_s (n)\rfloor$ and runs in $O(n(1 + \ell /s^2)/p + (s^2 + \ell) \log_s (n))$ time assuming $p$ tensor core units. Equivalently, the algorithm performs $O(n/s^2)$ multiplications of square matrices of size s.

cs.DC↗

arXiv2022

Identifying and Exploiting Sparse Branch Correlations for Optimizing Branch Prediction

Branch prediction is arguably one of the most important speculative mechanisms within a high-performance processor architecture. A common approach to improve branch prediction accuracy is to employ lengthy history records of previously seen branch directions to capture distant correlations between branches. The larger the history, the richer the information that the predictor can exploit for discovering predictive patterns. However, without appropriate filtering, such an approach may also heavily disorganize the predictor's internal mechanisms, leading to diminishing returns. This paper studies a fundamental control-flow property: the sparsity in the correlation between branches and recent history. First, we show that sparse branch correlations exist in standard applications and, more importantly, such correlations can be computed efficiently using sparse modeling methods. Second, we introduce a sparsity-aware branch prediction mechanism that can compactly encode and store sparse models to unlock essential performance opportunities. We evaluated our approach for various design parameters demonstrating MPKI improvements of up to 42% (2.3% on average) with 2KB of additional storage overhead. Our circuit-level evaluation of the design showed that it can operate within accepted branch prediction latencies, and under reasonable power and area limitations.

cs.AR↗

arXiv2021

Branch Prediction as a Reinforcement Learning Problem: Why, How and Case Studies

Recent years have seen stagnating improvements to branch predictor (BP) efficacy and a dearth of fresh ideas in branch predictor design, calling for fresh thinking in this area. This paper argues that looking at BP from the viewpoint of Reinforcement Learning (RL) facilitates systematic reasoning about, and exploration of, BP designs. We describe how to apply the RL formulation to branch predictors, show that existing predictors can be succinctly expressed in this formulation, and study two RL-based variants of conventional BPs.

cs.LG↗

arXiv2020

Team voyTECH: User Activity Modeling with Boosting Trees

This paper describes our winning solution for the ECML-PKDD ChAT Discovery Challenge 2020. We show that whether or not a Twitch user has subscribed to a channel can be well predicted by modeling user activity with boosting trees. We introduce the connection between target-encodings and boosting trees in the context of high cardinality categoricals and find that modeling user activity is more powerful then direct modeling of content when encoded properly and combined with a suitable optimization approach.

cs.LG↗

arXiv2016

A Randomized Algorithm for Approximating the Log Determinant of a Symmetric Positive Definite Matrix

We introduce a novel algorithm for approximating the logarithm of the determinant of a symmetric positive definite (SPD) matrix. The algorithm is randomized and approximates the traces of a small number of matrix powers of a specially constructed matrix, using the method of Avron and Toledo~\cite{AT11}. From a theoretical perspective, we present additive and relative error bounds for our algorithm. Our additive error bound works for any SPD matrix, whereas our relative error bound works for SPD matrices whose eigenvalues lie in the interval $(θ_1,1)$, with $0<θ_1<1$; the latter setting was proposed in~\cite{icml2015_hana15}. From an empirical perspective, we demonstrate that a C++ implementation of our algorithm can approximate the logarithm of the determinant of large matrices very accurately in a matter of seconds.

cs.DS↗

arXiv2014

Randomized Dimensionality Reduction for k-means Clustering

We study the topic of dimensionality reduction for $k$-means clustering. Dimensionality reduction encompasses the union of two approaches: \emph{feature selection} and \emph{feature extraction}. A feature selection based algorithm for $k$-means clustering selects a small subset of the input features and then applies $k$-means clustering on the selected features. A feature extraction based algorithm for $k$-means clustering constructs a small set of new artificial features and then applies $k$-means clustering on the constructed features. Despite the significance of $k$-means clustering as well as the wealth of heuristic methods addressing it, provably accurate feature selection methods for $k$-means clustering are not known. On the other hand, two provably accurate feature extraction methods for $k$-means clustering are known in the literature; one is based on random projections and the other is based on the singular value decomposition (SVD). This paper makes further progress towards a better understanding of dimensionality reduction for $k$-means clustering. Namely, we present the first provably accurate feature selection method for $k$-means clustering and, in addition, we present two feature extraction methods. The first feature extraction method is based on random projections and it improves upon the existing results in terms of time complexity and number of features needed to be extracted. The second feature extraction method is based on fast approximate SVD factorizations and it also improves upon the existing results in terms of time complexity. The proposed algorithms are randomized and provide constant-factor approximation guarantees with respect to the optimal $k$-means objective value.

cs.DS↗

arXiv2014

Approximate Matrix Multiplication with Application to Linear Embeddings

In this paper, we study the problem of approximately computing the product of two real matrices. In particular, we analyze a dimensionality-reduction-based approximation algorithm due to Sarlos [1], introducing the notion of nuclear rank as the ratio of the nuclear norm over the spectral norm. The presented bound has improved dependence with respect to the approximation error (as compared to previous approaches), whereas the subspace -- on which we project the input matrices -- has dimensions proportional to the maximum of their nuclear rank and it is independent of the input dimensions. In addition, we provide an application of this result to linear low-dimensional embeddings. Namely, we show that any Euclidean point-set with bounded nuclear rank is amenable to projection onto number of dimensions that is independent of the input dimensionality, while achieving additive error guarantees.

math.ST↗

arXiv2014

Non-uniform Feature Sampling for Decision Tree Ensembles

We study the effectiveness of non-uniform randomized feature selection in decision tree classification. We experimentally evaluate two feature selection methodologies, based on information extracted from the provided dataset: $(i)$ \emph{leverage scores-based} and $(ii)$ \emph{norm-based} feature selection. Experimental evaluation of the proposed feature selection techniques indicate that such approaches might be more effective compared to naive uniform feature selection and moreover having comparable performance to the random forest algorithm [3]

stat.ML↗

arXiv2014

Randomized Block Kaczmarz Method with Projection for Solving Least Squares

The Kaczmarz method is an iterative method for solving overcomplete linear systems of equations Ax=b. The randomized version of the Kaczmarz method put forth by Strohmer and Vershynin iteratively projects onto a randomly chosen solution space given by a single row of the matrix A and converges exponentially in expectation to the solution of a consistent system. In this paper we analyze two block versions of the method each with a randomized projection, that converge in expectation to the least squares solution of inconsistent systems. Our approach utilizes a paving of the matrix A to guarantee exponential convergence, and suggests that paving yields a significant improvement in performance in certain regimes. The proposed method is an extension of the block Kaczmarz method analyzed by Needell and Tropp and the Randomized Extended Kaczmarz method of Zouzias and Freris. The contribution is thus two-fold; unlike the standard Kaczmarz method, our methods converge to the least-squares solution of inconsistent systems, and by using appropriate blocks of the matrix this convergence can be significantly accelerated. Numerical experiments suggest that the proposed algorithm can indeed lead to advantages in practice.

math.NA↗

arXiv2012

Hidden cliques and the certification of the restricted isometry property

Compressed sensing is a technique for finding sparse solutions to underdetermined linear systems. This technique relies on properties of the sensing matrix such as the restricted isometry property. Sensing matrices that satisfy this property with optimal parameters are mainly obtained via probabilistic arguments. Deciding whether a given matrix satisfies the restricted isometry property is a non-trivial computational problem. Indeed, we show in this paper that restricted isometry parameters cannot be approximated in polynomial time within any constant factor under the assumption that the hidden clique problem is hard. Moreover, on the positive side we propose an improvement on the brute-force enumeration algorithm for checking the restricted isometry property.

cs.CC↗

arXiv2012

Efficient Dimensionality Reduction for Canonical Correlation Analysis

We present a fast algorithm for approximate Canonical Correlation Analysis (CCA). Given a pair of tall-and-thin matrices, the proposed algorithm first employs a randomized dimensionality reduction transform to reduce the size of the input matrices, and then applies any CCA algorithm to the new pair of matrices. The algorithm computes an approximate CCA to the original pair of matrices with provable guarantees, while requiring asymptotically less operations than the state-of-the-art exact algorithms.

cs.DS↗

arXiv2012

Randomized Extended Kaczmarz for Solving Least-Squares

We present a randomized iterative algorithm that exponentially converges in expectation to the minimum Euclidean norm least squares solution of a given linear system of equations. The expected number of arithmetic operations required to obtain an estimate of given accuracy is proportional to the square condition number of the system multiplied by the number of non-zeros entries of the input matrix. The proposed algorithm is an extension of the randomized Kaczmarz method that was analyzed by Strohmer and Vershynin.

math.NA↗

arXiv2012

A Matrix Hyperbolic Cosine Algorithm and Applications

In this paper, we generalize Spencer's hyperbolic cosine algorithm to the matrix-valued setting. We apply the proposed algorithm to several problems by analyzing its computational efficiency under two special cases of matrices; one in which the matrices have a group structure and an other in which they have rank-one. As an application of the former case, we present a deterministic algorithm that, given the multiplication table of a finite group of size $n$, it constructs an expanding Cayley graph of logarithmic degree in near-optimal O(n^2 log^3 n) time. For the latter case, we present a fast deterministic algorithm for spectral sparsification of positive semi-definite matrices, which implies an improved deterministic algorithm for spectral graph sparsification of dense graphs. In addition, we give an elementary connection between spectral sparsification of positive semi-definite matrices and element-wise matrix sparsification. As a consequence, we obtain improved element-wise sparsification algorithms for diagonally dominant-like matrices.

cs.DS↗