arXiv subjects

Andrei Chertkov

Publications and source records attributed to Andrei Chertkov.

15 recordsLinked to original sources

arXiv2026

Global Optimization of Atomic Clusters via Physically-Constrained Tensor Train Decomposition

The global optimization of atomic clusters represents a fundamental challenge in computational chemistry and materials science due to the exponential growth of local minima with system size (i.e., the curse of dimensionality). We introduce a novel framework that overcomes this limitation by exploiting the low-rank structure of potential energy surfaces through Tensor Train (TT) decomposition. Our approach combines two complementary TT-based strategies: the algebraic TTOpt method, which utilizes maximum volume sampling, and the probabilistic PROTES method, which employs generative sampling. A key innovation is the development of physically-constrained encoding schemes that incorporate molecular constraints directly into the discretization process. We demonstrate the efficacy of our method by identifying global minima of Lennard-Jones clusters containing up to 45 atoms. Furthermore, we establish its practical applicability to real-world systems by optimizing 20-atom carbon clusters using a machine-learned Moment Tensor Potential, achieving geometries consistent with quantum-accurate simulations. This work establishes TT-decomposition as a powerful tool for molecular structure prediction and provides a general framework adaptable to a wide range of high-dimensional optimization problems in computational material science.

math.OC↗

arXiv2025

Low-rank surrogate modeling and stochastic zero-order optimization for training of neural networks with black-box layers

The growing demand for energy-efficient, high-performance AI systems has led to increased attention on alternative computing platforms (e.g., photonic, neuromorphic) due to their potential to accelerate learning and inference. However, integrating such physical components into deep learning pipelines remains challenging, as physical devices often offer limited expressiveness, and their non-differentiable nature renders on-device backpropagation difficult or infeasible. This motivates the development of hybrid architectures that combine digital neural networks with reconfigurable physical layers, which effectively behave as black boxes. In this work, we present a framework for the end-to-end training of such hybrid networks. This framework integrates stochastic zeroth-order optimization for updating the physical layer's internal parameters with a dynamic low-rank surrogate model that enables gradient propagation through the physical layer. A key component of our approach is the implicit projector-splitting integrator algorithm, which updates the lightweight surrogate model after each forward pass with minimal hardware queries, thereby avoiding costly full matrix reconstruction. We demonstrate our method across diverse deep learning tasks, including: computer vision, audio classification, and language modeling. Notably, across all modalities, the proposed approach achieves near-digital baseline accuracy and consistently enables effective end-to-end training of hybrid models incorporating various non-differentiable physical components (spatial light modulators, microring resonators, and Mach-Zehnder interferometers). This work bridges hardware-aware deep learning and gradient-free optimization, thereby offering a practical pathway for integrating non-differentiable physical components into scalable, end-to-end trainable AI systems.

cs.LG↗

arXiv2025

High-dimensional Optimization with Low Rank Tensor Sampling and Local Search

We present a novel method called TESALOCS (TEnsor SAmpling and LOCal Search) for multidimensional optimization, combining the strengths of gradient-free discrete methods and gradient-based approaches. The discrete optimization in our method is based on low-rank tensor techniques, which, thanks to their low-parameter representation, enable efficient optimization of high-dimensional problems. For the second part, i.e., local search, any effective gradient-based method can be used, whether existing (such as quasi-Newton methods) or any other developed in the future. Our approach addresses the limitations of gradient-based methods, such as getting stuck in local optima; the limitations of discrete methods, which cannot be directly applied to continuous functions; and limitations of gradient-free methods that require large computational budgets. Note that we are not limited to a single type of low-rank tensor decomposition for discrete optimization, but for illustrative purposes, we consider a specific efficient low-rank tensor train decomposition. For 20 challenging 100-dimensional functions, we demonstrate that our method can significantly outperform results obtained with gradient-based methods like Conjugate Gradient, BFGS, SLSQP, and other methods, improving them by orders of magnitude with the same computing budget.

math.OC↗

arXiv2024

Faster Language Models with Better Multi-Token Prediction Using Tensor Decomposition

We propose a new model for multi-token prediction in transformers, aiming to enhance sampling efficiency without compromising accuracy. Motivated by recent work that predicts the probabilities of subsequent tokens using multiple heads, we connect this approach to rank-$1$ canonical tensor decomposition. By generalizing it to a rank-$r$ canonical probability decomposition, we develop an improved model that predicts multiple tokens simultaneously. This model can also be interpreted as a mixture of experts, allowing us to leverage successful techniques from that domain for efficient and robust training. Importantly, the overall overhead for training and sampling remains low. Our method demonstrates significant improvements in inference speed for both text and code generation tasks, proving particularly beneficial within the self-speculative decoding paradigm. It maintains its effectiveness across various model sizes and training epochs, highlighting its robustness and scalability.

cs.LG↗

arXiv2024

Black-Box Approximation and Optimization with Hierarchical Tucker Decomposition

We develop a new method HTBB for the multidimensional black-box approximation and gradient-free optimization, which is based on the low-rank hierarchical Tucker decomposition with the use of the MaxVol indices selection procedure. Numerical experiments for 14 complex model problems demonstrate the robustness of the proposed method for dimensions up to 1000, while it shows significantly more accurate results than classical gradient-free optimization methods, as well as approximation and optimization methods based on the popular tensor train decomposition, which represents a simpler case of a tensor network.

cs.LG↗

arXiv2023

Fast gradient-free activation maximization for neurons in spiking neural networks

Elements of neural networks, both biological and artificial, can be described by their selectivity for specific cognitive features. Understanding these features is important for understanding the inner workings of neural networks. For a living system, such as a neuron, whose response to a stimulus is unknown and not differentiable, the only way to reveal these features is through a feedback loop that exposes it to a large set of different stimuli. The properties of these stimuli should be varied iteratively in order to maximize the neuronal response. To utilize this feedback loop for a biological neural network, it is important to run it quickly and efficiently in order to reach the stimuli that maximizes certain neurons' activation with the least number of iterations possible. Here we present a framework with an efficient design for such a loop. We successfully tested it on an artificial spiking neural network (SNN), which is a model that simulates the asynchronous spiking activity of neurons in living brains. Our optimization method for activation maximization is based on the low-rank Tensor Train decomposition of the discrete activation function. The optimization space is the latent parameter space of images generated by SN-GAN or VQ-VAE generative models. To our knowledge, this is the first time that effective AM has been applied to SNNs. We track changes in the optimal stimuli for artificial neurons during training and show that highly selective neurons can form already in the early epochs of training and in the early layers of a convolutional spiking network. This formation of refined optimal stimuli is associated with an increase in classification accuracy. Some neurons, especially in the deeper layers, may gradually change the concepts they are selective for during learning, potentially explaining their importance for model performance.

cs.NE↗

arXiv2023

Tensor Train Decomposition for Adversarial Attacks on Computer Vision Models

Deep neural networks (DNNs) are widely used today, but they are vulnerable to adversarial attacks. To develop effective methods of defense, it is important to understand the potential weak spots of DNNs. Often attacks are organized taking into account the architecture of models (white-box approach) and based on gradient methods, but for real-world DNNs this approach in most cases is impossible. At the same time, several gradient-free optimization algorithms are used to attack black-box models. However, classical methods are often ineffective in the multidimensional case. To organize black-box attacks for computer vision models, in this work, we propose the use of an optimizer based on the low-rank tensor train (TT) format, which has gained popularity in various practical multidimensional applications in recent years. Combined with the attribution of the target image, which is built by the auxiliary (white-box) model, the TT-based optimization method makes it possible to organize an effective black-box attack by small perturbation of pixels in the target image. The superiority of the proposed approach over three popular baselines is demonstrated for seven modern DNNs on the ImageNet dataset.

math.NA↗

arXiv2023

Translate your gibberish: black-box adversarial attack on machine translation systems

Neural networks are deployed widely in natural language processing tasks on the industrial scale, and perhaps the most often they are used as compounds of automatic machine translation systems. In this work, we present a simple approach to fool state-of-the-art machine translation tools in the task of translation from Russian to English and vice versa. Using a novel black-box gradient-free tensor-based optimizer, we show that many online translation tools, such as Google, DeepL, and Yandex, may both produce wrong or offensive translations for nonsensical adversarial input queries and refuse to translate seemingly benign input phrases. This vulnerability may interfere with understanding a new language and simply worsen the user's experience while using machine translation systems, and, hence, additional improvements of these tools are required to establish better translation.

cs.CL↗

arXiv2023

PROTES: Probabilistic Optimization with Tensor Sampling

We developed a new method PROTES for black-box optimization, which is based on the probabilistic sampling from a probability density function given in the low-parametric tensor train format. We tested it on complex multidimensional arrays and discretized multivariable functions taken, among others, from real-world applications, including unconstrained binary optimization and optimal control problems, for which the possible number of elements is up to $2^{100}$. In numerical experiments, both on analytic model functions and on complex problems, PROTES outperforms existing popular discrete optimization methods (Particle Swarm Optimization, Covariance Matrix Adaptation, Differential Evolution, and others).

math.NA↗

arXiv2022

Understanding DDPM Latent Codes Through Optimal Transport

Diffusion models have recently outperformed alternative approaches to model the distribution of natural images, such as GANs. Such diffusion models allow for deterministic sampling via the probability flow ODE, giving rise to a latent space and an encoder map. While having important practical applications, such as estimation of the likelihood, the theoretical properties of this map are not yet fully understood. In the present work, we partially address this question for the popular case of the VP SDE (DDPM) approach. We show that, perhaps surprisingly, the DDPM encoder map coincides with the optimal transport map for common distributions; we support this claim theoretically and by extensive numerical experiments.

stat.ML↗

arXiv2022

Optimization of Functions Given in the Tensor Train Format

Tensor train (TT) format is a common approach for computationally efficient work with multidimensional arrays, vectors, matrices, and discretized functions in a wide range of applications, including computational mathematics and machine learning. In this work, we propose a new algorithm for TT-tensor optimization, which leads to very accurate approximations for the minimum and maximum tensor element. The method consists in sequential tensor multiplications of the TT-cores with an intelligent selection of candidates for the optimum. We propose the probabilistic interpretation of the method, and make estimates on its complexity and convergence. We perform extensive numerical experiments with random tensors and various multivariable benchmark functions with the number of input dimensions up to $100$. Our approach generates a solution close to the exact optimum for all model problems, while the running time is no more than $50$ seconds on a regular laptop.

math.NA↗

arXiv2022

TTOpt: A Maximum Volume Quantized Tensor Train-based Optimization and its Application to Reinforcement Learning

We present a novel procedure for optimization based on the combination of efficient quantized tensor train representation and a generalized maximum matrix volume principle. We demonstrate the applicability of the new Tensor Train Optimizer (TTOpt) method for various tasks, ranging from minimization of multidimensional functions to reinforcement learning. Our algorithm compares favorably to popular evolutionary-based methods and outperforms them by the number of function evaluations or execution time, often by a significant margin.

cs.LG↗

arXiv2022

Black box approximation in the tensor train format initialized by ANOVA decomposition

Surrogate models can reduce computational costs for multivariable functions with an unknown internal structure (black boxes). In a discrete formulation, surrogate modeling is equivalent to restoring a multidimensional array (tensor) from a small part of its elements. The alternating least squares (ALS) algorithm in the tensor train (TT) format is a widely used approach to effectively solve this problem in the case of non-adaptive tensor recovery from a given training set (i.e., tensor completion problem). TT-ALS allows obtaining a low-parametric representation of the tensor, which is free from the curse of dimensionality and can be used for fast computation of the values at arbitrary tensor indices or efficient implementation of algebra operations with the black box (integration, etc.). However, to obtain high accuracy in the presence of restrictions on the size of the train data, a good choice of initial approximation is essential. In this work, we construct the ANOVA representation in the TT-format and use it as an initial approximation for the TT-ALS algorithm. The performed numerical computations for a number of multidimensional model problems, including the parametric partial differential equation, demonstrate a significant advantage of our approach for the commonly used random initial approximation. For all considered model problems we obtained an increase in accuracy by at least an order of magnitude with the same number of requests to the black box. The proposed approach is very general and can be applied in a wide class of real-world surrogate modeling and machine learning problems.

math.NA↗

arXiv2022

Are Quantum Computers Practical Yet? A Case for Feature Selection in Recommender Systems using Tensor Networks

Collaborative filtering models generally perform better than content-based filtering models and do not require careful feature engineering. However, in the cold-start scenario collaborative information may be scarce or even unavailable, whereas the content information may be abundant, but also noisy and expensive to acquire. Thus, selection of particular features that improve cold-start recommendations becomes an important and non-trivial task. In the recent approach by Nembrini et al., the feature selection is driven by the correlational compatibility between collaborative and content-based models. The problem is formulated as a Quadratic Unconstrained Binary Optimization (QUBO) which, due to its NP-hard complexity, is solved using Quantum Annealing on a quantum computer provided by D-Wave. Inspired by the reported results, we contend the idea that current quantum annealers are superior for this problem and instead focus on classical algorithms. In particular, we tackle QUBO via TTOpt, a recently proposed black-box optimizer based on tensor networks and multilinear algebra. We show the computational feasibility of this method for large problems with thousands of features, and empirically demonstrate that the solutions found are comparable to the ones obtained with D-Wave across all examined datasets.

cs.IR↗

arXiv2021

Solution of the Fokker-Planck equation by cross approximation method in the tensor train format

We propose the novel numerical scheme for solution of the multidimensional Fokker-Planck equation, which is based on the Chebyshev interpolation and the spectral differentiation techniques as well as low rank tensor approximations, namely, the tensor train decomposition and the multidimensional cross approximation method, which in combination makes it possible to drastically reduce the number of degrees of freedom required to maintain accuracy as dimensionality increases. We demonstrate the effectiveness of the proposed approach on a number of multidimensional problems, including Ornstein-Uhlenbeck process and the dumbbell model. The developed computationally efficient solver can be used in a wide range of practically significant problems, including density estimation in machine learning applications.

math.NA↗