Tensor decomposition faces fundamental challenges in analyzing high-dimensional data, where traditional methods based on reconstruction and fixed-rank constraints often fail to capture semantically meaningful structures. This paper introduces a no-rank tensor decomposition framework grounded in metric learning, which replaces reconstruction objectives with a discriminative, similarity-based optimization. The proposed approach learns data-driven embeddings by optimizing a triplet loss with diversity and uniformity regularization, creating a feature space where distance directly reflects semantic similarity. We provide theoretical guarantees for the framework's convergence and establish bounds on its metric properties. Evaluations across diverse domains -- including face recognition (LFW, Olivetti), brain connectivity analysis (ABIDE), and simulated data (galaxy morphology, crystal structures) -- demonstrate that our method outperforms baseline techniques, including PCA, t-SNE, UMAP, and tensor decomposition baselines (CP and Tucker). Results show substantial improvements in clustering metrics (Silhouette Score, Davies-Bouldin Index, Calinski-Harabasz Index, Separation Ratio, Adjusted Rand Index, Normalized Mutual Information) and reveal a fundamental trade-off: while metric learning optimizes global class separation, it deliberately transforms local geometry to align with semantic relationships. Crucially, our approach achieves superior performance with smaller training datasets compared to transformer-based methods, offering an efficient alternative for domains with limited labeled data. This work establishes metric learning as a paradigm for tensor-based analysis, prioritizing semantic relevance over pixel-level fidelity while providing computational advantages in data-scarce scenarios.

Post-feedback comments: - I understand the spectral method depends on various problem-specific constants, but it's still not clear to me why its convergence rate is sub-optimal.

Comparing tensors and identifying their (dis)similar structures is fundamental in understanding the underlying phenomena for complex data. Tensor decomposition methods help analysts extract tensors' essential characteristics and aid in visual analytics for tensors. In contrast to dimensionality reduction (DR) methods designed only for analyzing a matrix (i.e., second-order tensor), existing tensor decomposition methods do not support flexible comparative analysis. To address this analysis limitation, we introduce a new tensor decomposition method, named tensor unified linear comparative analysis (TULCA), by extending its DR counterpart, ULCA, for tensor analysis. TULCA integrates discriminant analysis and contrastive learning schemes for tensor decomposition, enabling flexible comparison of tensors. We also introduce an effective method to visualize a core tensor extracted from TULCA into a set of 2D visualizations. We integrate TULCA's functionalities into a visual analytics interface to support analysts in interpreting and refining the TULCA results. We demonstrate the efficacy of TULCA and the visual analytics interface with computational evaluations and two case studies, including an analysis of log data collected from a supercomputer.

Main ideas of the submission The manuscript presents an approximation of nonnegative multi-way tensorial data (or high-order probability mass functions) based on structured energy function form that minimizes the Kullback-Leibler divergence. Comparing against other multilinear decomposition methods of nonnegative tensors, the proposal approach operates on multiplicative parameters under convex objective function and converges to a globally optimal solution. It also shows interesting connections with graphical models such as the high-order Boltzmann machines. Two optimization algorithms are developed, based upon gradient and natural gradient, respectively. The experiment shows that under the same number of parameters, the proposed approach yields smaller RMSEs than the other two baseline non-negative tensor decomposition methods.

Learning dynamic models from observed data has been a central issue in many scientific studies or engineering tasks. The usual setting is that data are collected sequentially from trajectories of some dynamical system operation. In quite a few modern scientific modeling tasks, however, it turns out that reliable sequential data are rather difficult to gather, whereas out-of-order snapshots are much easier to obtain. Examples include the modeling of galaxies, chronic diseases such Alzheimer's, or certain biological processes. Existing methods for learning dynamic model from non-sequence data are mostly based on Expectation-Maximization, which involves non-convex optimization and is thus hard to analyze. Inspired by recent advances in spectral learning methods, we propose to study this problem from a different perspective: moment matching and spectral decomposition. Under that framework, we identify reasonable assumptions on the generative process of non-sequence data, and propose learning algorithms based on the tensor decomposition method [2] to provably recover firstorder Markov models and hidden Markov models. To the best of our knowledge, this is the first formal guarantee on learning from non-sequence data. Preliminary simulation results confirm our theoretical findings.

As deep neural networks and the datasets used to train them get larger, the default approach to integrating them into research and commercial projects is to download a pre-trained model and fine tune it. But these models can have uncertain provenance, opening up the possibility that they embed hidden malicious behavior such as trojans or backdoors, where small changes to an input (triggers) can cause the model to produce incorrect outputs (e.g., to misclassify). This paper introduces a novel approach to backdoor detection that uses two tensor decomposition methods applied to network activations. This has a number of advantages relative to existing detection methods, including the ability to analyze multiple models at the same time, working across a wide variety of network architectures, making no assumptions about the nature of triggers used to alter network behavior, and being computationally efficient. We provide a detailed description of the detection pipeline along with results on models trained on the MNIST digit dataset, CIFAR-10 dataset, and two difficult datasets from NIST's TrojAI competition. These results show that our method detects backdoored networks more accurately and efficiently than current state-of-the-art methods.

Modern neural networks have revolutionized the fields of computer vision (CV) and Natural Language Processing (NLP). They are widely used for solving complex CV tasks and NLP tasks such as image classification, image generation, and machine translation. Most state-of-the-art neural networks are over-parameterized and require a high computational cost. One straightforward solution is to replace the layers of the networks with their low-rank tensor approximations using different tensor decomposition methods. This paper reviews six tensor decomposition methods and illustrates their ability to compress model parameters of convolutional neural networks (CNNs), recurrent neural networks (RNNs) and Transformers. The accuracy of some compressed models can be higher than the original versions. Evaluations indicate that tensor decompositions can achieve significant reductions in model size, run-time and energy consumption, and are well suited for implementing neural networks on edge devices.

Tensor decomposition methods (TDMs) have recently gained popularity as ways of performing inference for latent variable models [Anandkumar et al., 2014]. The interest in these methods is motivated by the fact that they come with theoretical global convergence guarantees in the limit of infinite data [Anandkumar et al., 2012, Arora et al., 2013]. However, a main limitation of these methods is that they lack natural methods for regularization or encouraging desired properties on the model parameters when the amount of data is limited. Previous works attempted to alleviate this drawback by modifying existing tensor decomposition methods to incorporate specific constraints, such as sparsity [Sun et al., 2015], or incorporate modeling assumptions, such as the existence of anchor words [Arora et al., 2013, Nguyen et al., 2014]. All of these works develop bespoke algorithms tailored to those constraints or assumptions. Furthermore, many of these methods impose hard constraints on the learned model, which may be detrimental as the size of the data grow--framed in the context of Bayesian intuition, when we have a lot of data, we want our methods to allow the evidence to overwhelm our priors. We introduce an alternative approach which can be applied to encourage any (differentiable) desired structure or properties on the model parameters, and which will only encourage this "prior" information when the data is insufficient. Specifically, we adopt the common view of Bayesian priors as representing "pseudo-observations" of artificial data which bias our learned model parameters towards our prior belief [Bishop, 2006]. We apply the tensor decomposition method of Anandkumar et al.

We present a novel nonnegative tensor decomposition method, called Legendre decomposition, which factorizes an input tensor into a multiplicative combination of parameters. Thanks to the well-developed theory of information geometry, the reconstructed tensor is unique and always minimizes the KL divergence from an input tensor. We empirically show that Legendre decomposition can more accurately reconstruct tensors than other nonnegative tensor decomposition methods.

We present a novel nonnegative tensor decomposition method, called Legendre decomposition, which factorizes an input tensor into a multiplicative combination of parameters. Thanks to the well-developed theory of information geometry, the reconstructed tensor is unique and always minimizes the KL divergence from an input tensor. We empirically show that Legendre decomposition can more accurately reconstruct tensors than other nonnegative tensor decomposition methods.