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We study transfer learning in the context of estimating piecewise-constant signals when source data, which may be relevant but disparate, are available in addition to the target data. We initially investigate transfer learning estimators that respectively employ $\ell_1$- and $\ell_0$-penalties for unisource data scenarios and then generalise these estimators to accommodate multisource data. To further reduce estimation errors, especially in scenarios where some sources significantly differ from the target, we introduce an informative source selection algorithm. We then examine these estimators with multisource selection and establish their minimax optimality under specific regularity conditions. It is worth emphasising that, unlike the prevalent narrative in the transfer learning literature that the performance is enhanced through large source sample sizes, our approaches leverage higher observation frequencies and accommodate diverse frequencies across multiple sources. Our theoretical findings are empirically validated through extensive numerical experiments, with the code available online, see //github.com/chrisfanwang/transferlearning

A finite element based computational scheme is developed and employed to assess a duality based variational approach to the solution of the linear heat and transport PDE in one space dimension and time, and the nonlinear system of ODEs of Euler for the rotation of a rigid body about a fixed point. The formulation turns initial-(boundary) value problems into degenerate elliptic boundary value problems in (space)-time domains representing the Euler-Lagrange equations of suitably designed dual functionals in each of the above problems. We demonstrate reasonable success in approximating solutions of this range of parabolic, hyperbolic, and ODE primal problems, which includes energy dissipation as well as conservation, by a unified dual strategy lending itself to a variational formulation. The scheme naturally associates a family of dual solutions to a unique primal solution; such `gauge invariance' is demonstrated in our computed solutions of the heat and transport equations, including the case of a transient dual solution corresponding to a steady primal solution of the heat equation. Primal evolution problems with causality are shown to be correctly approximated by non-causal dual problems.

Electrical circuits are present in a variety of technologies, making their design an important part of computer aided engineering. The growing number of tunable parameters that affect the final design leads to a need for new approaches of quantifying their impact. Machine learning may play a key role in this regard, however current approaches often make suboptimal use of existing knowledge about the system at hand. In terms of circuits, their description via modified nodal analysis is well-understood. This particular formulation leads to systems of differential-algebraic equations (DAEs) which bring with them a number of peculiarities, e.g. hidden constraints that the solution needs to fulfill. We aim to use the recently introduced dissection concept for DAEs that can decouple a given system into ordinary differential equations, only depending on differential variables, and purely algebraic equations that describe the relations between differential and algebraic variables. The idea then is to only learn the differential variables and reconstruct the algebraic ones using the relations from the decoupling. This approach guarantees that the algebraic constraints are fulfilled up to the accuracy of the nonlinear system solver, which represents the main benefit highlighted in this article.

A central challenge in the verification of quantum computers is benchmarking their performance as a whole and demonstrating their computational capabilities. In this work, we find a universal model of quantum computation, Bell sampling, that can be used for both of those tasks and thus provides an ideal stepping stone towards fault-tolerance. In Bell sampling, we measure two copies of a state prepared by a quantum circuit in the transversal Bell basis. We show that the Bell samples are classically intractable to produce and at the same time constitute what we call a circuit shadow: from the Bell samples we can efficiently extract information about the quantum circuit preparing the state, as well as diagnose circuit errors. In addition to known properties that can be efficiently extracted from Bell samples, we give two new and efficient protocols, a test for the depth of the circuit and an algorithm to estimate a lower bound to the number of T gates in the circuit. With some additional measurements, our algorithm learns a full description of states prepared by circuits with low T-count.

Research in high energy physics (HEP) requires huge amounts of computing and storage, putting strong constraints on the code speed and resource usage. To meet these requirements, a compiled high-performance language is typically used; while for physicists, who focus on the application when developing the code, better research productivity pleads for a high-level programming language. A popular approach consists of combining Python, used for the high-level interface, and C++, used for the computing intensive part of the code. A more convenient and efficient approach would be to use a language that provides both high-level programming and high-performance. The Julia programming language, developed at MIT especially to allow the use of a single language in research activities, has followed this path. In this paper the applicability of using the Julia language for HEP research is explored, covering the different aspects that are important for HEP code development: runtime performance, handling of large projects, interface with legacy code, distributed computing, training, and ease of programming. The study shows that the HEP community would benefit from a large scale adoption of this programming language. The HEP-specific foundation libraries that would need to be consolidated are identified

The elliptic curve discrete logarithm problem is of fundamental importance in public-key cryptography. It is in use for a long time. Moreover, it is an interesting challenge in computational mathematics. Its solution is supposed to provide interesting research directions. In this paper, we explore ways to solve the elliptic curve discrete logarithm problem. Our results are mostly computational. However, it seems, the methods that we develop and directions that we pursue can provide a potent attack on this problem. This work follows our earlier work, where we tried to solve this problem by finding a zero minor in a matrix over the same finite field on which the elliptic curve is defined. This paper is self-contained.

The spectral clustering algorithm is often used as a binary clustering method for unclassified data by applying the principal component analysis. To study theoretical properties of the algorithm, the assumption of conditional homoscedasticity is often supposed in existing studies. However, this assumption is restrictive and often unrealistic in practice. Therefore, in this paper, we consider the allometric extension model, that is, the directions of the first eigenvectors of two covariance matrices and the direction of the difference of two mean vectors coincide, and we provide a non-asymptotic bound of the error probability of the spectral clustering algorithm for the allometric extension model. As a byproduct of the result, we obtain the consistency of the clustering method in high-dimensional settings.

Inspired by biology's most sophisticated computer, the brain, neural networks constitute a profound reformulation of computational principles. Remarkably, analogous high-dimensional, highly-interconnected computational architectures also arise within information-processing molecular systems inside living cells, such as signal transduction cascades and genetic regulatory networks. Might neuromorphic collective modes be found more broadly in other physical and chemical processes, even those that ostensibly play non-information-processing roles such as protein synthesis, metabolism, or structural self-assembly? Here we examine nucleation during self-assembly of multicomponent structures, showing that high-dimensional patterns of concentrations can be discriminated and classified in a manner similar to neural network computation. Specifically, we design a set of 917 DNA tiles that can self-assemble in three alternative ways such that competitive nucleation depends sensitively on the extent of co-localization of high-concentration tiles within the three structures. The system was trained in-silico to classify a set of 18 grayscale 30 x 30 pixel images into three categories. Experimentally, fluorescence and atomic force microscopy monitoring during and after a 150-hour anneal established that all trained images were correctly classified, while a test set of image variations probed the robustness of the results. While slow compared to prior biochemical neural networks, our approach is surprisingly compact, robust, and scalable. This success suggests that ubiquitous physical phenomena, such as nucleation, may hold powerful information processing capabilities when scaled up as high-dimensional multicomponent systems.

Both centralized and decentralized approaches have shown excellent performance and great application value in federated learning (FL). However, current studies do not provide sufficient evidence to show which one performs better. Although from the optimization perspective, decentralized methods can approach the comparable convergence of centralized methods with less communication, its test performance has always been inefficient in empirical studies. To comprehensively explore their behaviors in FL, we study their excess risks, including the joint analysis of both optimization and generalization. We prove that on smooth non-convex objectives, 1) centralized FL (CFL) always generalizes better than decentralized FL (DFL); 2) from perspectives of the excess risk and test error in CFL, adopting partial participation is superior to full participation; and, 3) there is a necessary requirement for the topology in DFL to avoid performance collapse as the training scale increases. Based on some simple hardware metrics, we could evaluate which framework is better in practice. Extensive experiments are conducted on common setups in FL to validate that our theoretical analysis is contextually valid in practical scenarios.

Dataset Distillation (DD) is a prominent technique that encapsulates knowledge from a large-scale original dataset into a small synthetic dataset for efficient training. Meanwhile, Pre-trained Models (PTMs) function as knowledge repositories, containing extensive information from the original dataset. This naturally raises a question: Can PTMs effectively transfer knowledge to synthetic datasets, guiding DD accurately? To this end, we conduct preliminary experiments, confirming the contribution of PTMs to DD. Afterwards, we systematically study different options in PTMs, including initialization parameters, model architecture, training epoch and domain knowledge, revealing that: 1) Increasing model diversity enhances the performance of synthetic datasets; 2) Sub-optimal models can also assist in DD and outperform well-trained ones in certain cases; 3) Domain-specific PTMs are not mandatory for DD, but a reasonable domain match is crucial. Finally, by selecting optimal options, we significantly improve the cross-architecture generalization over baseline DD methods. We hope our work will facilitate researchers to develop better DD techniques. Our code is available at //github.com/yaolu-zjut/DDInterpreter.