Rank Decomposition Research Articles

Unscented recursive three-step filter based unbiased minimum-variance estimation for a class of nonlinear systems

For the direct feedthrough system where unknown input affects both process equation and measurement equation, an unscented recursive three-step filter based unbiased minimum-variance (URTSF-UMV) estimation algorithm is proposed. The algorithm is based on the unbiased minimum-variance (UMV) method and uses statistical linearisation technology to approximate the nonlinear system and the nonlinear measurement function to linear regression form. The estimation of unknown input is obtained from innovation through weighted least-squares (WLS) estimation. The approximate linear regression form allows us to process the nonlinear system with unknown input direct feedthrough by using the UMV state estimation framework, and derive the filter by minimising the trace of the state error covariance of the measurement update under the unbiased condition. The URTSF-UMV requires that the unknown input distribution matrix is full column rank. We use the full rank decomposition method to deal with the nonlinear system containing the unknown input rank-deficient distribution matrix. The effectiveness of the URTSF-UMV algorithm is demonstrated through an illustrative simulation example.

Determination of ballistic resistance model of finite thickness material based on Artificial Neural Network

Ballistic resistance behavior of finite thickness material is affected by factors of target thickness, projectile nose shape, and impact angle. The widely used ballistic limit velocity (BLV) model can only reflect the ballistic resistance behavior of a target in a single given impact scenario. The collected BLV from different impact scenarios can only partially reflect the effect of those factors on the ballistic resistance. A general model of BLV considering these factors simultaneously is desired.In this study, numerical impact models are first verified by a significant number of experiments. A guided random sampling method is proposed to collect numerical BLV data for training/validating usage. BLV data from experiments and simulations are then assembled by this method and subsequently trained by Artificial Neural Network (ANN). k-Folder Cross Validation (K-CV) technique designed for evaluating small sample training network is introduced to validate and test the network. The ANN-predicted results are decomposed by proper singular value decomposition methods. It is found that the Rank-1 decomposition results (with highest singular value) can fully reveal the decoupled relationship between BLV and its independent factors (MAPE < 6 %), with which analytical ballistic models are established. The results show that our proposed model can quantitatively describe the fully decoupled effect of target thickness, impact angle and projectile nose shape on the BLV with high accuracy.

On the bottleneck stability of rank decompositions of multi-parameter persistence modules

A significant part of modern topological data analysis is concerned with the design and study of algebraic invariants of poset representations—often referred to as persistence modules. One such invariant is the minimal rank decomposition, which encodes the ranks of all the structure morphisms of the persistence module by a single ordered pair of rectangle-decomposable modules, interpreted as a signed barcode. This signed barcode generalizes the concept of persistence barcode from one-parameter persistence to any number of parameters, raising the question of its bottleneck stability. We show in this paper that the minimal rank decomposition is not stable under the natural notion of signed bottleneck matching between signed barcodes. We remedy this by turning our focus to the rank exact decomposition, a related signed barcode induced by the minimal projective resolution of the module relative to the so-called rank exact structure, which we prove to be bottleneck stable under signed matchings. As part of our proof, we obtain two intermediate results of independent interest: we compute the global dimension of the rank exact structure on the category of finitely presentable multi-parameter persistence modules, and we prove a bottleneck stability result for hook-decomposable modules. We also give a bound for the size of the rank exact decomposition that is polynomial in the size of the usual minimal projective resolution, we prove a universality result for the dissimilarity function induced by the notion of signed matching, and we compute, in the two-parameter case, the global dimension of a different exact structure related to the upsets of the indexing poset. This set of results combines concepts from topological data analysis and from the representation theory of posets, and we believe is relevant to both areas.

On the classification of simple amenable $\mathrm{C}^{*}$-algebras with finite decomposition rank, II

We prove that every unital simple separable \mathrm{C}^{*} -algebra A with finite decomposition rank which satisfies the UCT has the property that A\otimes Q has generalized tracial rank at most one, where Q is the universal UHF-algebra. Consequently, A is classifiable in the sense of Elliott.

Modeling and Learning on High-Dimensional Matrix-Variate Sequences

We propose a new matrix factor model, named RaDFaM, which is strictly derived from the general rank decomposition and assumes a high-dimensional vector factor model structure for each basis vector. RaDFaM contributes a novel class of low-rank latent structures that trade off between signal intensity and dimension reduction from a tensor subspace perspective. Based on the intrinsic separable covariance structure of RaDFaM, for a collection of matrix-valued observations, we derive a new class of PCA variants for estimating loading matrices, and sequentially the latent factor matrices. The peak signal-to-noise ratio of RaDFaM is proved to be superior in the category of PCA-type estimators. We also establish an asymptotic theory including the consistency, convergence rates, and asymptotic distributions for components in the signal part. Numerically, we demonstrate the performance of RaDFaM in applications such as matrix reconstruction, supervised learning, and clustering, on uncorrelated and correlated data, respectively. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Finite approximation properties of C*-modules III

Abstract We introduce and study a notion of module nuclear dimension for a C * \mathrm{C}^{*} -algebra A which is a C * \mathrm{C}^{*} -module over another C * \mathrm{C}^{*} -algebra 𝔄 {\mathfrak{A}} with compatible actions. We show that the module nuclear dimension of A is zero if A is 𝔄 {\mathfrak{A}} -NF. The converse is shown to hold when 𝔄 {\mathfrak{A}} is a C ⁢ ( X ) {C(X)} -algebra with simple fibers, with X compact and totally disconnected. We also introduce a notion of module decomposition rank, and show that when 𝔄 {\mathfrak{A}} is unital and simple, if the module decomposition rank of A is finite then A is 𝔄 {\mathfrak{A}} -QD. We study the set 𝒯 𝔄 ⁢ ( A ) {\mathcal{T}_{\mathfrak{A}}(A)} of 𝔄 {\mathfrak{A}} -valued module traces on A and relate the Cuntz semigroup of A with lower semicontinuous affine functions on the set 𝒯 𝔄 ⁢ ( A ) {\mathcal{T}_{\mathfrak{A}}(A)} . Along the way, we also prove a module Choi–Effros lifting theorem. We give estimates of the module nuclear dimension for a class of examples.

Windowed Radon Transform and Tensor Rank-1 Decomposition for Adaptive Beamforming in Ultrafast Ultrasound.

Ultrafast ultrasound has recently emerged as an alternative to traditional focused ultrasound. By virtue of the low number of insonifications it requires, ultrafast ultrasound enables the imaging of the human body at potentially very high frame rates. However, unaccounted for speed-of-sound variations in the insonified medium often result in phase aberrations in the reconstructed images. The diagnosis capability of ultrafast ultrasound is thus ultimately impeded. Therefore, there is a strong need for adaptive beamforming methods that are resilient to speed-of-sound aberrations. Several of such techniques have been proposed recently but they often lack parallelizability or the ability to directly correct both transmit and receive phase aberrations. In this article, we introduce an adaptive beamforming method designed to address these shortcomings. To do so, we compute the windowed Radon transform of several complex radio-frequency images reconstructed using delay-and-sum. Then, we apply to the obtained local sinograms weighted tensor rank-1 decompositions and their results are eventually used to reconstruct a corrected image. We demonstrate using simulated and in-vitro data that our method is able to successfully recover aberration-free images and that it outperforms both coherent compounding and the recently introduced SVD beamformer. Finally, we validate the proposed beamforming technique on in-vivo data, resulting in a significant improvement of image quality compared to the two reference methods.

TCM-GPT: Efficient pre-training of large language models for domain adaptation in Traditional Chinese Medicine

Pre-training and fine-tuning have emerged as a promising paradigm across various natural language processing (NLP) tasks. The effectiveness of pretrained large language models (LLM) has witnessed further enhancement, holding potential for applications in the field of medicine, particularly in the context of Traditional Chinese Medicine (TCM). However, the application of these general models to specific domains often yields suboptimal results, primarily due to challenges like lack of domain knowledge, unique objectives, and computational efficiency. Furthermore, their effectiveness in specialized domains, such as Traditional Chinese Medicine, requires comprehensive evaluation.To address the above issues, we propose a novel domain specific TCMDA (TCM Domain Adaptation) approach, efficient pre-training with domain-specific corpus. Specifically, we first construct a large TCM-specific corpus, TCM-Corpus-1B, by identifying domain keywords and retrieving from general corpus. Then, our TCMDA leverages the LoRA which freezes the pretrained model’s weights and uses rank decomposition matrices to efficiently train specific dense layers for pre-training and fine-tuning, efficiently aligning the model with TCM-related tasks, namely TCM-GPT-7B. We further conducted extensive experiments on two TCM tasks, including TCM examination and TCM diagnosis. TCM-GPT-7B archived the best performance across both datasets, outperforming other models by relative increments of 17% and 12% in accuracy, respectively. To the best of our knowledge, our study represents the pioneering validation of domain adaptation of a large language model with 7 billion parameters in TCM domain. We will release both TCM-Corpus-1B and TCM-GPT-7B model once accepted to facilitate interdisciplinary development in TCM and NLP, serving as the foundation for further study.

Enhancing GANs with MMD Neural Architecture Search, PMish Activation Function and Adaptive Rank Decomposition

Enhancing GANs with MMD Neural Architecture Search, PMish Activation Function and Adaptive Rank Decomposition

Matrix Factorization For Augmented Deep Learning Model Generation

This paper aims to address two significant challenges of Deep Learning(DL) model generation, high computational cost involved during the training phase and data interpretability from high dimensional data. The computational cost is curbed using centralized as well as distributed matrix factorization technique known as CUR decomposition. CUR decomposition has the advantage of reducing the dimensions of the dataset without data transformation and minimal information loss. Though Singular Value De-composition(SVD) is the best matrix decomposition technique with least reconstruction error, it transforms the data and hence the original data cannot be interpreted from the decomposed matrices while applying DL technique. So CUR decomposition is leveraged to reduce the number of features in high-dimensional data while preserving the essential information. Extensive experimental analysis is conducted to evaluate and compare the performance of CUR and SVD techniques based on reconstruction error and time complexity in centralized as well as distributed setting. The results exhibit that CUR outperforms SVD in terms of computational efficiency, especially in a distributed setting where the decomposition time for CUR was a fraction compared to SVD. At the same time DL models generated from this reduced dataset exhibit comparable accuracy with the original high dimensional dataset as well as reduced dataset obtained through SVD. This work also enhances the accessibility of the CUR algorithm to users, for which an application is developed that enables users to execute the algorithm on any dataset. The user-friendly interface of the application facilitates the upload of datasets, configuration of desired decomposition rank, and execution of the algorithm with a single click.

Efficient contingency analysis of power systems using linear power flow with generalized warm-start compensation

Contingency analysis (CA) is fundamental to power system planning and operation. The compensation method and linear power flow (LPF) models are frequently used in CA to boost efficiency. However, the conventional compensation scheme fails to adapt to modern LPF models with novel structures. To bridge this gap, this paper proposes a generalized warm-start compensation algorithm (GWSCA) for LPFs. To efficiently calculate the inversion of a perturbed coefficient matrix in LPF models, a generalized decomposition method is devised based on the matrix inverse lemma and full rank decomposition. A warm-start strategy is developed to improve the computational accuracy by using the precontingency operating point of a power grid. The proposed GWSCA is incorporated with an LPF model based on logarithmic transformation of voltage magnitudes to implement fast and accurate CA. Simulation results show that GWSCA outperforms the conventional algorithms in terms of computational efficiency and accuracy.

The tensor of the exact circle: reconstructing geometry

Developing a theory for quantum gravity is one of the big open questions in theoretical high-energy physics. Recently, a tensor model approach has been considered that treats tensors as the generators of commutative non-associative algebras, which might be an appropriate interpretation of the canonical tensor model. In this approach, the non-associative algebra is assumed to be a low-energy description of the so-called associative closure, which gives the full description of spacetime including the high-energy modes. In the previous work it has been shown how to (re)construct a topological space with a measure on it, and one of the prominent examples that was used to develop the framework was the exact circle. In this work we will further investigate this example, and show that it is possible to reconstruct the full Riemannian geometry by reconstructing the metric tensor. Furthermore, it is demonstrated how diffeomorphisms behave in this formalism, firstly by considering a specific class of diffeomorphisms of the circle, namely the ellipses, and subsequently by performing an explicit diffeomorphism to ‘smoothen’ sets of points generated by the tensor rank decomposition.

Hyperspectral Anomaly Detection with Auto-Encoder and Independent Target

As an unsupervised data representation neural network, auto-encoder (AE) has shown great potential in denoising, dimensionality reduction, and data reconstruction. Many AE-based background (BKG) modeling methods have been developed for hyperspectral anomaly detection (HAD). However, their performance is subject to their unbiased reconstruction of BKG and target pixels. This article presents a rather different low rank and sparse matrix decomposition (LRaSMD) method based on AE, named auto-encoder and independent target (AE-IT), for hyperspectral anomaly detection. First, the encoder weight matrix, obtained by a designed AE network, is utilized to construct a projector for generating a low-rank component in the encoder subspace. By adaptively and reasonably determining the number of neurons in the latent layer, the designed AE-based method can promote the reconstruction of BKG. Second, to ensure independence and representativeness, the component in the encoder orthogonal subspace is made into a sphere and followed by finding of unsupervised targets to construct an anomaly space. In order to mitigate the influence of noise on anomaly detection, sparse cardinality (SC) constraint is enforced on the component in the anomaly space for obtaining the sparse anomaly component. Finally, anomaly detector is constructed by combining Mahalanobi distance and multi-components, which include encoder component and sparse anomaly component, to detect anomalies. The experimental results demonstrate that AE-IT performs competitively compared to the LRaSMD-based models and AE-based approaches.

Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models

The quest for a consistent theory for quantum gravity is one of the most challenging problems in theoretical high-energy physics. An often-used approach is to describe the gravitational degrees of freedom by the metric tensor or related variables, and finding a way to quantise this. In the canonical tensor model, the gravitational degrees of freedom are encoded in a tensorial quantity $P_{abc}$, and this quantity is subsequently quantised. This makes the quantisation much more straightforward mathematically, but the interpretation of this tensor as a spacetime is less evident. In this work we take a first step towards fully understanding the relationship to spacetime. By considering $P_{abc}$ as the generator of an algebra of functions, we first describe how we can recover the topology and the measure of a compact Riemannian manifold. Using the tensor rank decomposition, we then generalise this principle in order to have a well-defined notion of the topology and geometry for a large class of tensors $P_{abc}$. We provide some examples of the emergence of a topology and measure of both exact and perturbed Riemannian manifolds, and of a purely algebraically-defined space called the semi-local circle.

Low rank and sparse decomposition based on extended $${LL}_{p}$$ norm

Low rank and sparse decomposition based on extended $${LL}_{p}$$ norm

Through the Wall Radar Imaging via Kronecker-structured Huber-type RPCA

The detection of multiple targets in an enclosed scene, from its outside, is a challenging topic of research addressed by Through-the-Wall Radar Imaging (TWRI). Traditionally, TWRI methods operate in two steps: first the removal of wall clutter then followed by the recovery of targets positions. Recent approaches manage in parallel the processing of the wall and targets via low rank plus sparse matrix decomposition and obtain better performances. In this paper, we reformulate this precisely via a RPCA-type problem, where the sparse vector appears in a Kronecker product. We extend this approach by adding a robust distance with flexible structure to handle heterogeneous noise and outliers, which may appear in TWRI measurements. The resolution is achieved via the Alternating Direction Method of Multipliers (ADMM) and variable splitting to decouple the constraints. The removal of the front wall is achieved via a closed-form proximal evaluation and the recovery of targets is possible via a tailored Majorization–Minimization (MM) step. The analysis and validation of our method is carried out using Finite-Difference Time-Domain (FDTD) simulated data, which show the advantage of our method in detection performance over complex scenarios.

Face recognition technology based on low-rank joint sparse representation algorithm

With the improvement of computer computing power and the development of artificial intelligence technology, face recognition technology has made a major breakthrough, and has been popularized and applied in all areas of life. However, different face structure and pose will affect the accuracy of face recognition. To overcome the problem, a low rank joint sparse representation algorithm for face recognition is proposed. The low rank features of images are extracted by structure independent and pairwise rank decomposition methods. The extracted low rank features of the first level image and the low rank features of the second level image are sparsely represented. Finally, the residual rate model is used to classify the images, and the final result of face recognition is obtained. The experimental results show that the proposed SRP algorithm has a recognition accuracy of more than 92% in two different face recognition tests. In the mixed multi face pose test, PRS algorithm performs best in the recognition of 1, 2, 3, 4, and 5 multi face pose types, with recognition rates of 95%, 94%, 93%, 91%, and 90% respectively. The algorithm also has excellent recognition performance and robustness in identifying harsh environments such as fuzzy environments. The research content focuses on complex face recognition scenes, innovatively uses low rank to complete the extraction of face feature data, and combines sparse selection of classification features to improve the overall effect of face recognition. It has important reference value for improving the overall security and recognition rate of face recognition.

On Best Low Rank Approximation of Positive Definite Tensors

Tensors, or multi-indexed arrays, play an important role in machine learning and signal processing. These higher-order generalizations of matrices allow for preservation of higher-order structure in data, and low rank decompositions of tensors allow for recovery of underlying information. In many cases, the underlying tensor of interest is positive (semi)definite, i.e., the homogeneous polynomial associated to the tensor has nonnegative evaluation on all inputs. An archetypal problem is that one has a noisy measurement of some low rank signal tensor of interest. This measurement itself does not have low rank, so one must compute a best low rank approximation of the measured tensor to recover component information. As it turns out, the set of tensors of rank less than or equal to is, in general, not closed when , and, as a consequence, best low rank approximations can fail to exist. In the case when a best low rank approximation does not exist, near optimal low rank approximations suffer numerical issues and cannot be used to reliably approximate underlying component information. As a consequence, existence guarantees for best low rank tensor approximations are of great practical and theoretical interest. This article develops deterministic guarantees for the existence of best low rank approximations of tensors which are positive semidefinite. In particular, we show that the set of low rank positive semidefinite tensors is relatively closed as a subset of the set of positive semidefinite tensors. We use this fact to give a deterministic bound which may be used to guarantee the existence of a best low rank approximation of a noisy low rank positive semidefinite tensor. Furthermore, we show that this condition guarantees uniqueness of the canonical polyadic decomposition for best low rank approximations. In addition, for order three tensors, we prove that our bound is sharp and that it can be computed using semidefinite programming. The main results of this article are illustrated through numerical experiments, which show that our bound is highly predictive of numerical issues when attempting to recover underlying component information from a noisy low rank positive semidefinite tensor.

Quantum Physics: Illustration of Multidimensional Green Functions Using Julia Language

In this article, the numerical level representation of imaginary time Green's functions based on a low rank decomposition of the discrete Lehmann spectral representation (DLR) using an effective spectral density has been presented with illustrative efficiency. The DLR basis consists of a collection of exponentials chosen by interpolation decomposition to ensure stable and efficient recovery of imaginary time Green functions or Matsubara frequency samples (Kaye et al., 2022). This implementation to fit behavior of a low-temperature spinless free fermionics-electron gas quantum system requires much fewer degrees of freedom for the standard discretizations. The basic functions of DLR are explicit; carefully chosen to ensure a stable and accurate approach, simplifying standard operations. Importantly, the function can be explicitly transformed to the Matsubara frequency domain or obtained directly by interpolation on a Matsubara frequency grid.

Low Rank Tensor Decompositions and Approximations

AbstractThere exist linear relations among tensor entries of low rank tensors. These linear relations can be expressed by multi-linear polynomials, which are called generating polynomials. We use generating polynomials to compute tensor rank decompositions and low rank tensor approximations. We prove that this gives a quasi-optimal low rank tensor approximation if the given tensor is sufficiently close to a low rank one.