arXiv subjects

Boian Alexandrov

Publications and source records attributed to Boian Alexandrov.

At least 19 recordsLinked to original sources

arXiv2025

A Tensor Train-Based Isogeometric Solver for Large-Scale 3D Poisson Problems on Complex Geometries

We introduce a three-dimensional (3D) fully tensor train (TT)-assembled isogeometric analysis (IGA) framework, TT-IGA, for solving partial differential equations (PDEs) on complex geometries. Our method reformulates IGA discrete operators into TT format, enabling efficient compression and computation while retaining geometric flexibility and accuracy. Unlike previous low-rank approaches that typically rely on structured domains, our framework accommodates general 3D geometries through low-rank TT representations of both the geometry mapping and the PDE discretization. We demonstrate the effectiveness of the proposed TT-IGA framework on the 3D Poisson equation, achieving substantial reductions in memory usage and computational cost without compromising solution quality.

math.NA↗

arXiv2025

Matrix Factorization for Inferring Associations and Missing Links

Missing link prediction is a method for network analysis, with applications in recommender systems, biology, social sciences, cybersecurity, information retrieval, and Artificial Intelligence (AI) reasoning in Knowledge Graphs. Missing link prediction identifies unseen but potentially existing connections in a network by analyzing the observed patterns and relationships. In proliferation detection, this supports efforts to identify and characterize attempts by state and non-state actors to acquire nuclear weapons or associated technology - a notoriously challenging but vital mission for global security. Dimensionality reduction techniques like Non-Negative Matrix Factorization (NMF) and Logistic Matrix Factorization (LMF) are effective but require selection of the matrix rank parameter, that is, of the number of hidden features, k, to avoid over/under-fitting. We introduce novel Weighted (WNMFk), Boolean (BNMFk), and Recommender (RNMFk) matrix factorization methods, along with ensemble variants incorporating logistic factorization, for link prediction. Our methods integrate automatic model determination for rank estimation by evaluating stability and accuracy using a modified bootstrap methodology and uncertainty quantification (UQ), assessing prediction reliability under random perturbations. We incorporate Otsu threshold selection and k-means clustering for Boolean matrix factorization, comparing them to coordinate descent-based Boolean thresholding. Our experiments highlight the impact of rank k selection, evaluate model performance under varying test-set sizes, and demonstrate the benefits of UQ for reliable predictions using abstention. We validate our methods on three synthetic datasets (Boolean and uniformly distributed) and benchmark them against LMF and symmetric LMF (symLMF) on five real-world protein-protein interaction networks, showcasing an improved prediction performance.

cs.LG↗

arXiv2025

Topological Signatures of Adversaries in Multimodal Alignments

Multimodal Machine Learning systems, particularly those aligning text and image data like CLIP/BLIP models, have become increasingly prevalent, yet remain susceptible to adversarial attacks. While substantial research has addressed adversarial robustness in unimodal contexts, defense strategies for multimodal systems are underexplored. This work investigates the topological signatures that arise between image and text embeddings and shows how adversarial attacks disrupt their alignment, introducing distinctive signatures. We specifically leverage persistent homology and introduce two novel Topological-Contrastive losses based on Total Persistence and Multi-scale kernel methods to analyze the topological signatures introduced by adversarial perturbations. We observe a pattern of monotonic changes in the proposed topological losses emerging in a wide range of attacks on image-text alignments, as more adversarial samples are introduced in the data. By designing an algorithm to back-propagate these signatures to input samples, we are able to integrate these signatures into Maximum Mean Discrepancy tests, creating a novel class of tests that leverage topological signatures for better adversarial detection.

cs.LG↗

arXiv2024

HEAL: Hierarchical Embedding Alignment Loss for Improved Retrieval and Representation Learning

Retrieval-Augmented Generation (RAG) enhances Large Language Models (LLMs) by integrating external document retrieval to provide domain-specific or up-to-date knowledge. The effectiveness of RAG depends on the relevance of retrieved documents, which is influenced by the semantic alignment of embeddings with the domain's specialized content. Although full fine-tuning can align language models to specific domains, it is computationally intensive and demands substantial data. This paper introduces Hierarchical Embedding Alignment Loss (HEAL), a novel method that leverages hierarchical fuzzy clustering with matrix factorization within contrastive learning to efficiently align LLM embeddings with domain-specific content. HEAL computes level/depth-wise contrastive losses and incorporates hierarchical penalties to align embeddings with the underlying relationships in label hierarchies. This approach enhances retrieval relevance and document classification, effectively reducing hallucinations in LLM outputs. In our experiments, we benchmark and evaluate HEAL across diverse domains, including Healthcare, Material Science, Cyber-security, and Applied Maths.

cs.IR↗

arXiv2024

LoRID: Low-Rank Iterative Diffusion for Adversarial Purification

This work presents an information-theoretic examination of diffusion-based purification methods, the state-of-the-art adversarial defenses that utilize diffusion models to remove malicious perturbations in adversarial examples. By theoretically characterizing the inherent purification errors associated with the Markov-based diffusion purifications, we introduce LoRID, a novel Low-Rank Iterative Diffusion purification method designed to remove adversarial perturbation with low intrinsic purification errors. LoRID centers around a multi-stage purification process that leverages multiple rounds of diffusion-denoising loops at the early time-steps of the diffusion models, and the integration of Tucker decomposition, an extension of matrix factorization, to remove adversarial noise at high-noise regimes. Consequently, LoRID increases the effective diffusion time-steps and overcomes strong adversarial attacks, achieving superior robustness performance in CIFAR-10/100, CelebA-HQ, and ImageNet datasets under both white-box and black-box settings.

cs.LG↗

arXiv2024

LaFA: Latent Feature Attacks on Non-negative Matrix Factorization

As Machine Learning (ML) applications rapidly grow, concerns about adversarial attacks compromising their reliability have gained significant attention. One unsupervised ML method known for its resilience to such attacks is Non-negative Matrix Factorization (NMF), an algorithm that decomposes input data into lower-dimensional latent features. However, the introduction of powerful computational tools such as Pytorch enables the computation of gradients of the latent features with respect to the original data, raising concerns about NMF's reliability. Interestingly, naively deriving the adversarial loss for NMF as in the case of ML would result in the reconstruction loss, which can be shown theoretically to be an ineffective attacking objective. In this work, we introduce a novel class of attacks in NMF termed Latent Feature Attacks (LaFA), which aim to manipulate the latent features produced by the NMF process. Our method utilizes the Feature Error (FE) loss directly on the latent features. By employing FE loss, we generate perturbations in the original data that significantly affect the extracted latent features, revealing vulnerabilities akin to those found in other ML techniques. To handle large peak-memory overhead from gradient back-propagation in FE attacks, we develop a method based on implicit differentiation which enables their scaling to larger datasets. We validate NMF vulnerabilities and FE attacks effectiveness through extensive experiments on synthetic and real-world data.

cs.LG↗

arXiv2024

Tensor Train Low-rank Approximation (TT-LoRA): Democratizing AI with Accelerated LLMs

In recent years, Large Language Models (LLMs) have demonstrated remarkable capabilities across a wide range of natural language processing (NLP) tasks, such as question-answering, sentiment analysis, text summarization, and machine translation. However, the ever-growing complexity of LLMs demands immense computational resources, hindering the broader research and application of these models. To address this, various parameter-efficient fine-tuning strategies, such as Low-Rank Approximation (LoRA) and Adapters, have been developed. Despite their potential, these methods often face limitations in compressibility. Specifically, LoRA struggles to scale effectively with the increasing number of trainable parameters in modern large scale LLMs. Additionally, Low-Rank Economic Tensor-Train Adaptation (LoRETTA), which utilizes tensor train decomposition, has not yet achieved the level of compression necessary for fine-tuning very large scale models with limited resources. This paper introduces Tensor Train Low-Rank Approximation (TT-LoRA), a novel parameter-efficient fine-tuning (PEFT) approach that extends LoRETTA with optimized tensor train (TT) decomposition integration. By eliminating Adapters and traditional LoRA-based structures, TT-LoRA achieves greater model compression without compromising downstream task performance, along with reduced inference latency and computational overhead. We conduct an exhaustive parameter search to establish benchmarks that highlight the trade-off between model compression and performance. Our results demonstrate significant compression of LLMs while maintaining comparable performance to larger models, facilitating their deployment on resource-constraint platforms.

cs.LG↗

arXiv2024

Enhancing Code Translation in Language Models with Few-Shot Learning via Retrieval-Augmented Generation

The advent of large language models (LLMs) has significantly advanced the field of code translation, enabling automated translation between programming languages. However, these models often struggle with complex translation tasks due to inadequate contextual understanding. This paper introduces a novel approach that enhances code translation through Few-Shot Learning, augmented with retrieval-based techniques. By leveraging a repository of existing code translations, we dynamically retrieve the most relevant examples to guide the model in translating new code segments. Our method, based on Retrieval-Augmented Generation (RAG), substantially improves translation quality by providing contextual examples from which the model can learn in real-time. We selected RAG over traditional fine-tuning methods due to its ability to utilize existing codebases or a locally stored corpus of code, which allows for dynamic adaptation to diverse translation tasks without extensive retraining. Extensive experiments on diverse datasets with open LLM models such as Starcoder, Llama3-70B Instruct, CodeLlama-34B Instruct, Granite-34B Code Instruct, and Mixtral-8x22B, as well as commercial LLM models like GPT-3.5 Turbo and GPT-4o, demonstrate our approach's superiority over traditional zero-shot methods, especially in translating between Fortran and CPP. We also explored varying numbers of shots i.e. examples provided during inference, specifically 1, 2, and 3 shots and different embedding models for RAG, including Nomic-Embed, Starencoder, and CodeBERT, to assess the robustness and effectiveness of our approach.

cs.AI↗

arXiv2024

Tensor-Train WENO Scheme for Compressible Flows

In this study, we introduce a tensor-train (TT) finite difference WENO method for solving compressible Euler equations. In a step-by-step manner, the tensorization of the governing equations is demonstrated. We also introduce \emph{LF-cross} and \emph{WENO-cross} methods to compute numerical fluxes and the WENO reconstruction using the cross interpolation technique. A tensor-train approach is developed for boundary condition types commonly encountered in Computational Fluid Dynamics (CFD). The performance of the proposed WENO-TT solver is investigated in a rich set of numerical experiments. We demonstrate that the WENO-TT method achieves the theoretical $\text{5}^{\text{th}}$-order accuracy of the classical WENO scheme in smooth problems while successfully capturing complicated shock structures. In an effort to avoid the growth of TT ranks, we propose a dynamic method to estimate the TT approximation error that governs the ranks and overall truncation error of the WENO-TT scheme. Finally, we show that the traditional WENO scheme can be accelerated up to 1000 times in the TT format, and the memory requirements can be significantly decreased for low-rank problems, demonstrating the potential of tensor-train approach for future CFD application. This paper is the first study that develops a finite difference WENO scheme using the tensor-train approach for compressible flows. It is also the first comprehensive work that provides a detailed perspective into the relationship between rank, truncation error, and the TT approximation error for compressible WENO solvers.

math.NA↗

arXiv2024

Generating Hidden Markov Models from Process Models Through Nonnegative Tensor Factorization

Monitoring of industrial processes is a critical capability in industry and in government to ensure reliability of production cycles, quick emergency response, and national security. Process monitoring allows users to gauge the progress of an organization in an industrial process or predict the degradation or aging of machine parts in processes taking place at a remote location. Similar to many data science applications, we usually only have access to limited raw data, such as satellite imagery, short video clips, event logs, and signatures captured by a small set of sensors. To combat data scarcity, we leverage the knowledge of Subject Matter Experts (SMEs) who are familiar with the actions of interest. SMEs provide expert knowledge of the essential activities required for task completion and the resources necessary to carry out each of these activities. Various process mining techniques have been developed for this type of analysis; typically such approaches combine theoretical process models built based on domain expert insights with ad-hoc integration of available pieces of raw data. Here, we introduce a novel mathematically sound method that integrates theoretical process models (as proposed by SMEs) with interrelated minimal Hidden Markov Models (HMM), built via nonnegative tensor factorization. Our method consolidates: (a) theoretical process models, (b) HMMs, (c) coupled nonnegative matrix-tensor factorizations, and (d) custom model selection. To demonstrate our methodology and its abilities, we apply it on simple synthetic and real world process models.

cs.LG↗

arXiv2023

Electrical Grid Anomaly Detection via Tensor Decomposition

Supervisory Control and Data Acquisition (SCADA) systems often serve as the nervous system for substations within power grids. These systems facilitate real-time monitoring, data acquisition, control of equipment, and ensure smooth and efficient operation of the substation and its connected devices. Previous work has shown that dimensionality reduction-based approaches, such as Principal Component Analysis (PCA), can be used for accurate identification of anomalies in SCADA systems. While not specifically applied to SCADA, non-negative matrix factorization (NMF) has shown strong results at detecting anomalies in wireless sensor networks. These unsupervised approaches model the normal or expected behavior and detect the unseen types of attacks or anomalies by identifying the events that deviate from the expected behavior. These approaches; however, do not model the complex and multi-dimensional interactions that are naturally present in SCADA systems. Differently, non-negative tensor decomposition is a powerful unsupervised machine learning (ML) method that can model the complex and multi-faceted activity details of SCADA events. In this work, we novelly apply the tensor decomposition method Canonical Polyadic Alternating Poisson Regression (CP-APR) with a probabilistic framework, which has previously shown state-of-the-art anomaly detection results on cyber network data, to identify anomalies in SCADA systems. We showcase that the use of statistical behavior analysis of SCADA communication with tensor decomposition improves the specificity and accuracy of identifying anomalies in electrical grid systems. In our experiments, we model real-world SCADA system data collected from the electrical grid operated by Los Alamos National Laboratory (LANL) which provides transmission and distribution service through a partnership with Los Alamos County, and detect synthetically generated anomalies.

cs.LG↗

arXiv2023

Distributed Out-of-Memory NMF on CPU/GPU Architectures

We propose an efficient distributed out-of-memory implementation of the Non-negative Matrix Factorization (NMF) algorithm for heterogeneous high-performance-computing (HPC) systems. The proposed implementation is based on prior work on NMFk, which can perform automatic model selection and extract latent variables and patterns from data. In this work, we extend NMFk by adding support for dense and sparse matrix operation on multi-node, multi-GPU systems. The resulting algorithm is optimized for out-of-memory (OOM) problems where the memory required to factorize a given matrix is greater than the available GPU memory. Memory complexity is reduced by batching/tiling strategies, and sparse and dense matrix operations are significantly accelerated with GPU cores (or tensor cores when available). Input/Output (I/O) latency associated with batch copies between host and device is hidden using CUDA streams to overlap data transfers and compute asynchronously, and latency associated with collective communications (both intra-node and inter-node) is reduced using optimized NVIDIA Collective Communication Library NCCL based communicators. Benchmark results show significant improvement, from 32X to 76x speedup, with the new implementation using GPUs over the CPU-based NMFk. Good weak scaling was demonstrated on up to 4096 multi-GPU cluster nodes with approximately 25,000 GPUs when decomposing a dense 340 Terabyte-size matrix and an 11 Exabyte-size sparse matrix of density 10e-6.

cs.DC↗

arXiv2023

Robust Adversarial Defense by Tensor Factorization

As machine learning techniques become increasingly prevalent in data analysis, the threat of adversarial attacks has surged, necessitating robust defense mechanisms. Among these defenses, methods exploiting low-rank approximations for input data preprocessing and neural network (NN) parameter factorization have shown potential. Our work advances this field further by integrating the tensorization of input data with low-rank decomposition and tensorization of NN parameters to enhance adversarial defense. The proposed approach demonstrates significant defense capabilities, maintaining robust accuracy even when subjected to the strongest known auto-attacks. Evaluations against leading-edge robust performance benchmarks reveal that our results not only hold their ground against the best defensive methods available but also exceed all current defense strategies that rely on tensor factorizations. This study underscores the potential of integrating tensorization and low-rank decomposition as a robust defense against adversarial attacks in machine learning.

cs.LG↗

arXiv2022

Nonnegative Canonical Tensor Decomposition with Linear Constraints: nnCANDELINC

There is an emerging interest in tensor factorization applications in big-data analytics and machine learning. To speed up the factorization of extra-large datasets, organized in multidimensional arrays (aka tensors), easy to compute compression-based tensor representations, such as Tucker and Tensor Train formats, are used to approximate the initial large-tensor. Further, tensor factorization is used to extract latent features that can facilitate discoveries of new mechanisms and signatures hidden in the data, where the explainability of the latent features is of principal importance. Nonnegative tensor factorization extracts latent features that are naturally sparse and parts of the data, which makes them easily interpretable. However, to take into account available domain knowledge and subject matter expertise, additional constraints often need to be imposed, which lead us to Canonical decomposition with linear constraints (CANDELINC), a Canonical Polyadic Decomposition with rank deficient factors. In CANDELINC, Tucker compression is used as a pre-processing step, which leads to a larger residual error but to more explainable latent features. Here, we propose a nonnegative CANDELINC (nnCANDELINC) accomplished via a specific nonnegative Tucker decomposition; we refer to as minimal or canonical nonnegative Tucker. We derive several results required to understand the specificity of nnCANDELINC, focusing on the difficulties of preserving the nonnegative rank to its Tucker core and comparing the real-valued to the nonnegative case. Finally, we demonstrate nnCANDELINC performance on synthetic and real-world examples.

math.NA↗

arXiv2022

Distributed non-negative RESCAL with Automatic Model Selection for Exascale Data

With the boom in the development of computer hardware and software, social media, IoT platforms, and communications, there has been an exponential growth in the volume of data produced around the world. Among these data, relational datasets are growing in popularity as they provide unique insights regarding the evolution of communities and their interactions. Relational datasets are naturally non-negative, sparse, and extra-large. Relational data usually contain triples, (subject, relation, object), and are represented as graphs/multigraphs, called knowledge graphs, which need to be embedded into a low-dimensional dense vector space. Among various embedding models, RESCAL allows learning of relational data to extract the posterior distributions over the latent variables and to make predictions of missing relations. However, RESCAL is computationally demanding and requires a fast and distributed implementation to analyze extra-large real-world datasets. Here we introduce a distributed non-negative RESCAL algorithm for heterogeneous CPU/GPU architectures with automatic selection of the number of latent communities (model selection), called pyDRESCALk. We demonstrate the correctness of pyDRESCALk with real-world and large synthetic tensors, and the efficacy showing near-linear scaling that concurs with the theoretical complexities. Finally, pyDRESCALk determines the number of latent communities in an 11-terabyte dense and 9-exabyte sparse synthetic tensor.

cs.DC↗

arXiv2021

Factorization of Binary Matrices: Rank Relations, Uniqueness and Model Selection of Boolean Decomposition

The application of binary matrices are numerous. Representing a matrix as a mixture of a small collection of latent vectors via low-rank decomposition is often seen as an advantageous method to interpret and analyze data. In this work, we examine the factorizations of binary matrices using standard arithmetic (real and nonnegative) and logical operations (Boolean and $\mathbb{Z}_2$). We examine the relationships between the different ranks, and discuss when factorization is unique. In particular, we characterize when a Boolean factorization $X = W \land H$ has a unique $W$, a unique $H$ (for a fixed $W$), and when both $W$ and $H$ are unique, given a rank constraint. We introduce a method for robust Boolean model selection, called BMF$k$, and show on numerical examples that BMF$k$ not only accurately determines the correct number of Boolean latent features but reconstruct the pre-determined factors accurately.

math.NA↗

arXiv2021

Boolean Matrix Factorization via Nonnegative Auxiliary Optimization

A novel approach to Boolean matrix factorization (BMF) is presented. Instead of solving the BMF problem directly, this approach solves a nonnegative optimization problem with the constraint over an auxiliary matrix whose Boolean structure is identical to the initial Boolean data. Then the solution of the nonnegative auxiliary optimization problem is thresholded to provide a solution for the BMF problem. We provide the proofs for the equivalencies of the two solution spaces under the existence of an exact solution. Moreover, the nonincreasing property of the algorithm is also proven. Experiments on synthetic and real datasets are conducted to show the effectiveness and complexity of the algorithm compared to other current methods.

cs.DS↗

arXiv2020

Distributed Non-Negative Tensor Train Decomposition

The era of exascale computing opens new venues for innovations and discoveries in many scientific, engineering, and commercial fields. However, with the exaflops also come the extra-large high-dimensional data generated by high-performance computing. High-dimensional data is presented as multidimensional arrays, aka tensors. The presence of latent (not directly observable) structures in the tensor allows a unique representation and compression of the data by classical tensor factorization techniques. However, the classical tensor methods are not always stable or they can be exponential in their memory requirements, which makes them not suitable for high-dimensional tensors. Tensor train (TT) is a state-of-the-art tensor network introduced for factorization of high-dimensional tensors. TT transforms the initial high-dimensional tensor in a network of three-dimensional tensors that requires only a linear storage. Many real-world data, such as, density, temperature, population, probability, etc., are non-negative and for an easy interpretation, the algorithms preserving non-negativity are preferred. Here, we introduce a distributed non-negative tensor-train and demonstrate its scalability and the compression on synthetic and real-world big datasets.

cs.DC↗