Shakir Showkat Sofi

Shakir Showkat Sofi, Ivan Oseledets

The majority of real-world processes are spatiotemporal, and the data generated by them exhibits both spatial and temporal evolution. Weather is one of the most essential processes in this domain, and weather forecasting has become a crucial part of our daily routine. Weather data analysis is considered the most complex and challenging task. Although numerical weather prediction models are currently state-of-the-art, they are resource-intensive and time-consuming. Numerous studies have proposed time series-based models as a viable alternative to numerical forecasts. Recent research in the area of time series analysis indicates significant advancements, particularly regarding the use of state-space-based models (white box) and, more recently, the integration of machine learning and deep neural network-based models (black box). The most famous examples of such models are RNNs and transformers. These models have demonstrated remarkable results in the field of time-series analysis and have demonstrated effectiveness in modelling temporal correlations. It is crucial to capture both temporal and spatial correlations for a spatiotemporal process, as the values at nearby locations and time affect the values of a spatiotemporal process at a specific point. This self-contained paper explores various regional data-driven weather forecasting methods, i.e., forecasting over multiple latitude-longitude points (matrix-shaped spatial grid) to capture spatiotemporal correlations. The results showed that spatiotemporal prediction models reduced computational costs while improving accuracy. In particular, the proposed tensor train dynamic mode decomposition-based forecasting model has comparable accuracy to the state-of-the-art models without the need for training. We provide convincing numerical experiments to show that the proposed approach is practical.

Shakir Showkat Sofi, Charlotte Vermeylen, Lieven De Lathauwer

We present an algebraic algorithm for quantum state tomography that leverages measurements of certain observables to estimate structured entries of the underlying density matrix. Under low-rank assumptions, the remaining entries can be obtained solely using standard numerical linear algebra operations. The proposed algebraic matrix completion framework applies to a broad class of generic, low-rank mixed quantum states and, compared with state-of-the-art methods, is computationally efficient while providing deterministic recovery guarantees.

Shakir Showkat Sofi, Nadezhda Alsahanova

Solving segmentation tasks with topological priors proved to make fewer errors in fine-scale structures. In this work, we use topological priors both before and during the deep neural network training procedure. We compared the results of the two approaches with simple segmentation on various accuracy metrics and the Betti number error, which is directly related to topological correctness, and discovered that incorporating topological information into the classical UNet model performed significantly better. We conducted experiments on the ISBI EM segmentation dataset.

Shakir Showkat Sofi, Lieven De Lathauwer

Tensor completion is an extension of matrix completion aimed at recovering a multiway data tensor by leveraging a given subset of its entries (observations) and the pattern of observation. The low-rank assumption is key in establishing a relationship between the observed and unobserved entries of the tensor. The low-rank tensor completion problem is typically solved using numerical optimization techniques, where the rank information is used either implicitly (in the rank minimization approach) or explicitly (in the error minimization approach). Current theories concerning these techniques often study probabilistic recovery guarantees under conditions such as random uniform observations and incoherence requirements. However, if an observation pattern exhibits some low-rank structure that can be exploited, more efficient algorithms with deterministic recovery guarantees can be designed by leveraging this structure. This work shows how to use only standard linear algebra operations to compute the tensor train decomposition of a specific type of ``fiber-wise" observed tensor, where some of the fibers of a tensor (along a single specific mode) are either fully observed or entirely missing, unlike the usual entry-wise observations. From an application viewpoint, this setting is relevant when it is easier to sample or collect a multiway data tensor along a specific mode (e.g., temporal). The proposed completion method is fast and is guaranteed to work under reasonable deterministic conditions on the observation pattern. Through numerical experiments, we showcase interesting applications and use cases that illustrate the effectiveness of the proposed approach.

Shakir Showkat Sofi, Charlotte Vermeylen, Lieven De Lathauwer

Quantum state tomography (QST) is a fundamental technique for estimating the state of a quantum system from measured data and plays a crucial role in evaluating the performance of quantum devices. However, standard estimation methods become impractical due to the exponential growth of parameters in the state representation. In this work, we address this challenge by parameterizing the state using a low-rank block tensor train decomposition and demonstrate that our approach is both memory- and computationally efficient. This framework applies to a broad class of quantum states that can be well approximated by low-rank decompositions, including pure states, nearly pure states, and ground states of Hamiltonians.