Tensor CANDECOMP/PARAFAC (CP) decomposition has wide applications in statistical learning of latent variable models and in data mining. In this paper, we propose fast and randomized tensor CP decomposition algorithms based on sketching. We build on the idea of count sketches, but introduce many novel ideas which are unique to tensors. We develop novel methods for randomized computation of tensor contractions via FFTs, without explicitly forming the tensors. Such tensor contractions are encountered in decomposition methods such as tensor power iterations and alternating least squares. We also design novel colliding hashes for symmetric tensors to further save time in computing the sketches. We then combine these sketching ideas with existing whitening and tensor power iterative techniques to obtain the fastest algorithm on both sparse and dense tensors. The quality of approximation under our method does not depend on properties such as sparsity, uniformity of elements, etc. We apply the method for topic modeling and obtain competitive results.

Training large AI models such as deep learning recommendation systems and foundation language (or multi-modal) models costs massive GPUs and computing time. The high training cost has become only affordable to big tech companies, meanwhile also causing increasing concerns about the environmental impact. This paper presents CoMERA, a Computing- and Memory-Efficient training method via Rank-Adaptive tensor optimization. CoMERA achieves end-to-end rank-adaptive tensor-compressed training via a multi-objective optimization formulation, and improves the training to provide both a high compression ratio and excellent accuracy in the training process. Our optimized numerical computation (e.g., optimized tensorized embedding and tensor-vector contractions) and GPU implementation eliminate part of the run-time overhead in the tensorized training on GPU. This leads to, for the first time, $2-3\times$ speedup per training epoch compared with standard training. CoMERA also outperforms the recent GaLore in terms of both memory and computing efficiency. Specifically, CoMERA is $2\times$ faster per training epoch and $9\times$ more memory-efficient than GaLore on a tested six-encoder transformer with single-batch training. With further HPC optimization, CoMERA may significantly reduce the training cost of large language models.

Probabilistic inference is a fundamental task in modern machine learning. Recent advances in tensor network (TN) contraction algorithms have enabled the development of better exact inference methods. However, many common inference tasks in probabilistic graphical models (PGMs) still lack corresponding TN-based adaptations. In this work, we advance the connection between PGMs and TNs by formulating and implementing tensor-based solutions for the following inference tasks: (i) computing the partition function, (ii) computing the marginal probability of sets of variables in the model, (iii) determining the most likely assignment to a set of variables, and (iv) the same as (iii) but after having marginalized a different set of variables. We also present a generalized method for generating samples from a learned probability distribution. Our work is motivated by recent technical advances in the fields of quantum circuit simulation, quantum many-body physics, and statistical physics. Through an experimental evaluation, we demonstrate that the integration of these quantum technologies with a series of algorithms introduced in this study significantly improves the effectiveness of existing methods for solving probabilistic inference tasks.

We propose a new descriptor for local atomic environments, to be used in combination with machine learning models for the construction of interatomic potentials. The Local Atomic Tensors Trainable Expansion (LATTE) allows for the efficient construction of a variable number of many-body terms with learnable parameters, resulting in a descriptor that is efficient, expressive, and can be scaled to suit different accuracy and computational cost requirements. We compare this new descriptor to existing ones on several systems, showing it to be competitive with very fast potentials at one end of the spectrum, and extensible to an accuracy close to the state of the art.