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This article analyzes the algebraic structure of the set of all quantum channels and its subset consisting of quantum channels that have Holevo representation. The regularity of these semigroups under composition of mappings are analysed. It is also known that these sets are compact convex sets and, therefore, rich in geometry. An attempt is made to identify generalized invertible channels and also the idempotent channels. When channels are of the Holevo type, these two problems are fully studied in this article. The motivation behind this study is its applicability to the reversibility of channel transformations and recent developments in resource-destroying channels, which are idempotents. This is related to the coding-encoding problem in quantum information theory. Several examples are provided, with the main examples coming from pre-conditioner maps which assigns preconditioners to matrices, in numerical linear algebra.Thus the known pre-conditioner maps are viewd as a quantum-channel in finite dimentions.

Plug-and-Play Priors (PnP) is a well-known class of methods for solving inverse problems in computational imaging. PnP methods combine physical forward models with learned prior models specified as image denoisers. A common issue with the learned models is that of a performance drop when there is a distribution shift between the training and testing data. Test-time training (TTT) was recently proposed as a general strategy for improving the performance of learned models when training and testing data come from different distributions. In this paper, we propose PnP-TTT as a new method for overcoming distribution shifts in PnP. PnP-TTT uses deep equilibrium learning (DEQ) for optimizing a self-supervised loss at the fixed points of PnP iterations. PnP-TTT can be directly applied on a single test sample to improve the generalization of PnP. We show through simulations that given a sufficient number of measurements, PnP-TTT enables the use of image priors trained on natural images for image reconstruction in magnetic resonance imaging (MRI).

We study dual number symmetric matrices, dual complex Hermitian matrices and dual quaternion Hermitian matrices in a unified frame of dual Hermitian matrices. Suppose we have a ring, which can be the real field, the complex field, or the quaternion ring. Then an $n \times n$ dual Hermitian matrix has $n$ dual number eigenvalues. We define supplement matrices for a dual Hermitian matrix. Supplement matrices are Hermitian matrices in the original ring. The standard parts of the eigenvalues of that dual Hermitian matrix are the eigenvalues of the standard part Hermitian matrix in the original ring, while the dual parts of the eigenvalues of that dual Hermitian matrix are the eigenvalues of those {supplement} matrices. Hence, by apply any practical method for computing eigenvalues of Hermitian matrices in the original ring, we have a practical method for computing eigenvalues of a dual Hermitian matrix. We call this method the supplement matrix method. Applications to low rank approximation and generalized inverses of dual matrices, dual least squares problem and formation control are discussed. Numerical experiments are reported.

The capability to generate simulation-ready garment models from 3D shapes of clothed humans will significantly enhance the interpretability of captured geometry of real garments, as well as their faithful reproduction in the virtual world. This will have notable impact on fields like shape capture in social VR, and virtual try-on in the fashion industry. To align with the garment modeling process standardized by the fashion industry as well as cloth simulation softwares, it is required to recover 2D patterns. This involves an inverse garment design problem, which is the focus of our work here: Starting with an arbitrary target garment geometry, our system estimates an animatable garment model by automatically adjusting its corresponding 2D template pattern, along with the material parameters of the physics-based simulation (PBS). Built upon a differentiable cloth simulator, the optimization process is directed towards minimizing the deviation of the simulated garment shape from the target geometry. Moreover, our produced patterns meet manufacturing requirements such as left-to-right-symmetry, making them suited for reverse garment fabrication. We validate our approach on examples of different garment types, and show that our method faithfully reproduces both the draped garment shape and the sewing pattern.

Solutions to differential equations, which are used to model physical systems, are computed numerically by solving a set of discretized equations. This set of discretized equations is reduced to a large linear system, whose solution is typically found using an iterative solver. We start with an initial guess, $x_0$, and iterate the algorithm to obtain a sequence of solution vectors, $x_k$, which are approximations to the exact solution of the linear system, $x$. The iterative algorithm is said to converge to $x$, in the field of reals, if and only if $x_k$ converges to $x$ in the limit of $k \to \infty$. In this paper, we formally prove the asymptotic convergence of a particular class of iterative methods called the stationary iterative methods, in the Coq theorem prover. We formalize the necessary and sufficient conditions required for the iterative convergence, and extend this result to two classical iterative methods: the Gauss--Seidel method and the Jacobi method. For the Gauss--Seidel method, we also formalize a set of easily testable conditions for iterative convergence, called the Reich theorem, for a particular matrix structure, and apply this on a model problem of the one-dimensional heat equation. We also apply the main theorem of iterative convergence to prove convergence of the Jacobi method on the model problem.

The Skolem problem is a long-standing open problem in linear dynamical systems: can a linear recurrence sequence (LRS) ever reach 0 from a given initial configuration? Similarly, the positivity problem asks whether the LRS stays positive from an initial configuration. Deciding Skolem (or positivity) has been open for half a century: the best known decidability results are for LRS with special properties (e.g., low order recurrences). But these problems are easier for "uninitialized" variants, where the initial configuration is not fixed but can vary arbitrarily: checking if there is an initial configuration from which the LRS stays positive can be decided in polynomial time (Tiwari in 2004, Braverman in 2006). In this paper, we consider problems that lie between the initialized and uninitialized variants. More precisely, we ask if 0 (resp. negative numbers) can be avoided from every initial configuration in a neighborhood of a given initial configuration. This can be considered as a robust variant of the Skolem (resp. positivity) problem. We show that these problems lie at the frontier of decidability: if the neighbourhood is given as part of the input, then robust Skolem and robust positivity are Diophantine hard, i.e., solving either would entail major breakthroughs in Diophantine approximations, as happens for (non-robust) positivity. However, if one asks whether such a neighbourhood exists, then the problems turn out to be decidable with PSPACE complexity. Our techniques also allow us to tackle robustness for ultimate positivity, which asks whether there is a bound on the number of steps after which the LRS remains positive. There are two variants depending on whether we ask for a "uniform" bound on this number of steps. For the non-uniform variant, when the neighbourhood is open, the problem turns out to be tractable, even when the neighbourhood is given as input.

Support vector machines (SVMs) are widely used machine learning models (e.g., in remote sensing), with formulations for both classification and regression tasks. In the last years, with the advent of working quantum annealers, hybrid SVM models characterised by quantum training and classical execution have been introduced. These models have demonstrated comparable performance to their classical counterparts. However, they are limited in the training set size due to the restricted connectivity of the current quantum annealers. Hence, to take advantage of large datasets (like those related to Earth observation), a strategy is required. In the classical domain, local SVMs, namely, SVMs trained on the data samples selected by a k-nearest neighbors model, have already proven successful. Here, the local application of quantum-trained SVM models is proposed and empirically assessed. In particular, this approach allows overcoming the constraints on the training set size of the quantum-trained models while enhancing their performance. In practice, the FaLK-SVM method, designed for efficient local SVMs, has been combined with quantum-trained SVM models for binary and multiclass classification. In addition, for comparison, FaLK-SVM has been interfaced for the first time with a classical single-step multiclass SVM model (CS SVM). Concerning the empirical evaluation, D-Wave's quantum annealers and real-world datasets taken from the remote sensing domain have been employed. The results have shown the effectiveness and scalability of the proposed approach, but also its practical applicability in a real-world large-scale scenario.

Vitrimer is a new class of sustainable polymers with the ability of self-healing through rearrangement of dynamic covalent adaptive networks. However, a limited choice of constituent molecules restricts their property space, prohibiting full realization of their potential applications. Through a combination of molecular dynamics (MD) simulations and machine learning (ML), particularly a novel graph variational autoencoder (VAE) model, we establish a method for generating novel vitrimers and guide their inverse design based on desired glass transition temperature (Tg). We build the first vitrimer dataset of one million and calculate Tg on 8,424 of them by high-throughput MD simulations calibrated by a Gaussian process model. The proposed VAE employs dual graph encoders and a latent dimension overlapping scheme which allows for individual representation of multi-component vitrimers. By constructing a continuous latent space containing necessary information of vitrimers, we demonstrate high accuracy and efficiency of our framework in discovering novel vitrimers with desirable Tg beyond the training regime. The proposed vitrimers with reasonable synthesizability cover a wide range of Tg and broaden the potential widespread usage of vitrimeric materials.

This study investigates a strongly-coupled system of partial differential equations (PDE) governing heat transfer in a copper rod, longitudinal vibrations, and total charge accumulation at electrodes within a magnetizable piezoelectric beam. Conducted within the transmission line framework, the analysis reveals profound interactions between traveling electromagnetic and mechanical waves in magnetizable piezoelectric beams, despite disparities in their velocities. Findings suggest that in the open-loop scenario, the interaction of heat and beam dynamics lacks exponential stability solely considering thermal effects. To confront this challenge, two types of boundary-type state feedback controllers are proposed: (i) employing static feedback controllers entirely and (ii) adopting a hybrid approach wherein the electrical controller dynamically enhances system dynamics. In both cases, solutions of the PDE systems demonstrate exponential stability through meticulously formulated Lyapunov functions with diverse multipliers. The proposed proof technique establishes a robust foundation for demonstrating the exponential stability of Finite-Difference-based model reductions as the discretization parameter approaches zero.

While it is nearly effortless for humans to quickly assess the perceptual similarity between two images, the underlying processes are thought to be quite complex. Despite this, the most widely used perceptual metrics today, such as PSNR and SSIM, are simple, shallow functions, and fail to account for many nuances of human perception. Recently, the deep learning community has found that features of the VGG network trained on the ImageNet classification task has been remarkably useful as a training loss for image synthesis. But how perceptual are these so-called "perceptual losses"? What elements are critical for their success? To answer these questions, we introduce a new Full Reference Image Quality Assessment (FR-IQA) dataset of perceptual human judgments, orders of magnitude larger than previous datasets. We systematically evaluate deep features across different architectures and tasks and compare them with classic metrics. We find that deep features outperform all previous metrics by huge margins. More surprisingly, this result is not restricted to ImageNet-trained VGG features, but holds across different deep architectures and levels of supervision (supervised, self-supervised, or even unsupervised). Our results suggest that perceptual similarity is an emergent property shared across deep visual representations.