We study the optimal linear prediction of a random function, assuming it takes values in an infinite dimensional Hilbert space. We begin by characterizing the mean square prediction error (MSPE) associated with a linear predictor and discussing the minimal achievable MSPE. This analysis reveals that, in general, there are multiple non-unique linear predictors that minimize the MSPE, and even if a unique solution exists, consistently estimating it from finite samples is generally impossible. Nevertheless, we can define asymptotically optimal linear operators whose empirical MSPEs approach the minimal achievable level as the sample size increases. We show that, interestingly, standard post-dimension reduction estimators, which have been widely used in the literature, attain such asymptotic optimality under minimal conditions.
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Data depth functions have been intensively studied for normed vector spaces. However, a discussion on depth functions on data where one specific data structure cannot be presupposed is lacking. In this article, we introduce a notion of depth functions for data types that are not given in statistical standard data formats and therefore we do not have one specific data structure. We call such data in general non-standard data. To achieve this, we represent the data via formal concept analysis which leads to a unified data representation. Besides introducing depth functions for non-standard data using formal concept analysis, we give a systematic basis by introducing structural properties. Furthermore, we embed the generalised Tukey depth into our concept of data depth and analyse it using the introduced structural properties. Thus, this article provides the mathematical formalisation of centrality and outlyingness for non-standard data and therefore increases the spaces centrality is currently discussed. In particular, it gives a basis to define further depth functions and statistical inference methods for non-standard data.
This work proposes the extended functional tensor train (EFTT) format for compressing and working with multivariate functions on tensor product domains. Our compression algorithm combines tensorized Chebyshev interpolation with a low-rank approximation algorithm that is entirely based on function evaluations. Compared to existing methods based on the functional tensor train format, the adaptivity of our approach often results in reducing the required storage, sometimes considerably, while achieving the same accuracy. In particular, we reduce the number of function evaluations required to achieve a prescribed accuracy by up to over 96% compared to the algorithm from [Gorodetsky, Karaman and Marzouk, Comput. Methods Appl. Mech. Eng., 347 (2019)].
Interpolation of data on non-Euclidean spaces is an active research area fostered by its numerous applications. This work considers the Hermite interpolation problem: finding a sufficiently smooth manifold curve that interpolates a collection of data points on a Riemannian manifold while matching a prescribed derivative at each point. We propose a novel procedure relying on the general concept of retractions to solve this problem on a large class of manifolds, including those for which computing the Riemannian exponential or logarithmic maps is not straightforward, such as the manifold of fixed-rank matrices. We analyze the well-posedness of the method by introducing and showing the existence of retraction-convex sets, a generalization of geodesically convex sets. We extend to the manifold setting a classical result on the asymptotic interpolation error of Hermite interpolation. We finally illustrate these results and the effectiveness of the method with numerical experiments on the manifold of fixed-rank matrices and the Stiefel manifold of matrices with orthonormal columns.
The ability to extract material parameters of perovskite from quantitative experimental analysis is essential for rational design of photovoltaic and optoelectronic applications. However, the difficulty of this analysis increases significantly with the complexity of the theoretical model and the number of material parameters for perovskite. Here we use Gaussian process to develop an analysis platform that can extract up to 8 fundamental material parameters of an organometallic perovskite semiconductor from a transient photoluminescence experiment, based on a complex full physics model that includes drift-diffusion of carriers and dynamic defect occupation. An example study of thermal degradation reveals that changes in doping concentration and carrier mobility dominate, while the defect energy level remains nearly unchanged. This platform can be conveniently applied to other experiments or to combinations of experiments, accelerating materials discovery and optimization of semiconductor materials for photovoltaics and other applications.
Randomized matrix algorithms have become workhorse tools in scientific computing and machine learning. To use these algorithms safely in applications, they should be coupled with posterior error estimates to assess the quality of the output. To meet this need, this paper proposes two diagnostics: a leave-one-out error estimator for randomized low-rank approximations and a jackknife resampling method to estimate the variance of the output of a randomized matrix computation. Both of these diagnostics are rapid to compute for randomized low-rank approximation algorithms such as the randomized SVD and randomized Nystr\"om approximation, and they provide useful information that can be used to assess the quality of the computed output and guide algorithmic parameter choices.
We present Surjective Sequential Neural Likelihood (SSNL) estimation, a novel method for simulation-based inference in models where the evaluation of the likelihood function is not tractable and only a simulator that can generate synthetic data is available. SSNL fits a dimensionality-reducing surjective normalizing flow model and uses it as a surrogate likelihood function which allows for conventional Bayesian inference using either Markov chain Monte Carlo methods or variational inference. By embedding the data in a low-dimensional space, SSNL solves several issues previous likelihood-based methods had when applied to high-dimensional data sets that, for instance, contain non-informative data dimensions or lie along a lower-dimensional manifold. We evaluate SSNL on a wide variety of experiments and show that it generally outperforms contemporary methods used in simulation-based inference, for instance, on a challenging real-world example from astrophysics which models the magnetic field strength of the sun using a solar dynamo model.
The Knowledge Till rho CONGEST model is a variant of the classical CONGEST model of distributed computing in which each vertex v has initial knowledge of the radius-rho ball centered at v. The most commonly studied variants of the CONGEST model are KT0 CONGEST in which nodes initially know nothing about their neighbors and KT1 CONGEST in which nodes initially know the IDs of all their neighbors. It has been shown that having access to neighbors' IDs (as in the KT1 CONGEST model) can substantially reduce the message complexity of algorithms for fundamental problems such as BROADCAST and MST. For example, King, Kutten, and Thorup (PODC 2015) show how to construct an MST using just Otilde(n) messages in the KT1 CONGEST model, whereas there is an Omega(m) message lower bound for MST in the KT0 CONGEST model. Building on this result, Gmyr and Pandurangen (DISC 2018) present a family of distributed randomized algorithms for various global problems that exhibit a trade-off between message and round complexity. These algorithms are based on constructing a sparse, spanning subgraph called a danner. Specifically, given a graph G and any delta in [0,1], their algorithm constructs (with high probability) a danner that has diameter Otilde(D + n^{1-delta}) and Otilde(min{m,n^{1+delta}}) edges in Otilde(n^{1-delta}) rounds while using Otilde(min{m,n^{1+\delta}}) messages, where n, m, and D are the number of nodes, edges, and the diameter of G, respectively. In the main result of this paper, we show that if we assume the KT2 CONGEST model, it is possible to substantially improve the time-message trade-off in constructing a danner. Specifically, we show in the KT2 CONGEST model, how to construct a danner that has diameter Otilde(D + n^{1-2delta}) and Otilde(min{m,n^{1+delta}}) edges in Otilde(n^{1-2delta}) rounds while using Otilde(min{m,n^{1+\delta}}) messages for any delta in [0,1/2].
We study the problem of selecting $k$ experiments from a larger candidate pool, where the goal is to maximize mutual information (MI) between the selected subset and the underlying parameters. Finding the exact solution is to this combinatorial optimization problem is computationally costly, not only due to the complexity of the combinatorial search but also the difficulty of evaluating MI in nonlinear/non-Gaussian settings. We propose greedy approaches based on new computationally inexpensive lower bounds for MI, constructed via log-Sobolev inequalities. We demonstrate that our method outperforms random selection strategies, Gaussian approximations, and nested Monte Carlo (NMC) estimators of MI in various settings, including optimal design for nonlinear models with non-additive noise.
This paper delves into a nonparametric estimation approach for the interaction function within diffusion-type particle system models. We introduce two estimation methods based upon an empirical risk minimization. Our study encompasses an analysis of the stochastic and approximation errors associated with both procedures, along with an examination of certain minimax lower bounds. In particular, we show that there is a natural metric under which the corresponding minimax estimation error of the interaction function converges to zero with parametric rate. This result is rather suprising given complexity of the underlying estimation problem and rather large classes of interaction functions for which the above parametric rate holds.
Brain simulation builds dynamical models to mimic the structure and functions of the brain, while brain-inspired computing (BIC) develops intelligent systems by learning from the structure and functions of the brain. The two fields are intertwined and should share a common programming framework to facilitate each other's development. However, none of the existing software in the fields can achieve this goal, because traditional brain simulators lack differentiability for training, while existing deep learning (DL) frameworks fail to capture the biophysical realism and complexity of brain dynamics. In this paper, we introduce BrainPy, a differentiable brain simulator developed using JAX and XLA, with the aim of bridging the gap between brain simulation and BIC. BrainPy expands upon the functionalities of JAX, a powerful AI framework, by introducing complete capabilities for flexible, efficient, and scalable brain simulation. It offers a range of sparse and event-driven operators for efficient and scalable brain simulation, an abstraction for managing the intricacies of synaptic computations, a modular and flexible interface for constructing multi-scale brain models, and an object-oriented just-in-time compilation approach to handle the memory-intensive nature of brain dynamics. We showcase the efficiency and scalability of BrainPy on benchmark tasks, highlight its differentiable simulation for biologically plausible spiking models, and discuss its potential to support research at the intersection of brain simulation and BIC.
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