In this document, some elements of the theory and algorithmics corresponding to the existence and computability of approximate joint eigenpairs for finite collections of matrices with applications to model order reduction, are presented. More specifically, given a finite collection $X_1,\ldots,X_d$ of Hermitian matrices in $\mathbb{C}^{n\times n}$, a positive integer $r\ll n$, and a collection of complex numbers $\hat{x}_{j,k}\in \mathbb{C}$ for $1\leq j\leq d$, $1\leq k\leq r$. First, we study the computability of a set of $r$ vectors $w_1,\ldots,w_r\in \mathbb{C}^{n}$, such that $w_k=\arg\min_{w\in \mathbb{C}^n}\sum_{j=1}^d\|X_jw-\hat{x}_{j,k} w\|^2$ for each $1\leq k \leq r$, then we present a model order reduction procedure based on the truncated joint approximate eigenbases computed with the aforementioned techniques. Some prototypical algorithms together with some numerical examples are presented as well.
相關內容
We study the expressibility and learnability of convex optimization solution functions and their multi-layer architectural extension. The main results are: \emph{(1)} the class of solution functions of linear programming (LP) and quadratic programming (QP) is a universal approximant for the $C^k$ smooth model class or some restricted Sobolev space, and we characterize the rate-distortion, \emph{(2)} the approximation power is investigated through a viewpoint of regression error, where information about the target function is provided in terms of data observations, \emph{(3)} compositionality in the form of a deep architecture with optimization as a layer is shown to reconstruct some basic functions used in numerical analysis without error, which implies that \emph{(4)} a substantial reduction in rate-distortion can be achieved with a universal network architecture, and \emph{(5)} we discuss the statistical bounds of empirical covering numbers for LP/QP, as well as a generic optimization problem (possibly nonconvex) by exploiting tame geometry. Our results provide the \emph{first rigorous analysis of the approximation and learning-theoretic properties of solution functions} with implications for algorithmic design and performance guarantees.
In this paper, we propose a new approach to learned optimization. As common in the literature, we represent the computation of the update step of the optimizer with a neural network. The parameters of the optimizer are then learned on a set of training optimization tasks, in order to perform minimisation efficiently. Our main innovation is to propose a new neural network architecture for the learned optimizer inspired by the classic BFGS algorithm. As in BFGS, we estimate a preconditioning matrix as a sum of rank-one updates but use a transformer-based neural network to predict these updates jointly with the step length and direction. In contrast to several recent learned optimization approaches, our formulation allows for conditioning across different dimensions of the parameter space of the target problem while remaining applicable to optimization tasks of variable dimensionality without retraining. We demonstrate the advantages of our approach on a benchmark composed of objective functions traditionally used for evaluation of optimization algorithms, as well as on the real world-task of physics-based reconstruction of articulated 3D human motion.
Error correction in automatic speech recognition (ASR) aims to correct those incorrect words in sentences generated by ASR models. Since recent ASR models usually have low word error rate (WER), to avoid affecting originally correct tokens, error correction models should only modify incorrect words, and therefore detecting incorrect words is important for error correction. Previous works on error correction either implicitly detect error words through target-source attention or CTC (connectionist temporal classification) loss, or explicitly locate specific deletion/substitution/insertion errors. However, implicit error detection does not provide clear signal about which tokens are incorrect and explicit error detection suffers from low detection accuracy. In this paper, we propose SoftCorrect with a soft error detection mechanism to avoid the limitations of both explicit and implicit error detection. Specifically, we first detect whether a token is correct or not through a probability produced by a dedicatedly designed language model, and then design a constrained CTC loss that only duplicates the detected incorrect tokens to let the decoder focus on the correction of error tokens. Compared with implicit error detection with CTC loss, SoftCorrect provides explicit signal about which words are incorrect and thus does not need to duplicate every token but only incorrect tokens; compared with explicit error detection, SoftCorrect does not detect specific deletion/substitution/insertion errors but just leaves it to CTC loss. Experiments on AISHELL-1 and Aidatatang datasets show that SoftCorrect achieves 26.1% and 9.4% CER reduction respectively, outperforming previous works by a large margin, while still enjoying fast speed of parallel generation.
Image copy detection (ICD) aims to determine whether a query image is an edited copy of any image from a reference set. Currently, there are very limited public benchmarks for ICD, while all overlook a critical challenge in real-world applications, i.e., the distraction from hard negative queries. Specifically, some queries are not edited copies but are inherently similar to some reference images. These hard negative queries are easily false recognized as edited copies, significantly compromising the ICD accuracy. This observation motivates us to build the first ICD benchmark featuring this characteristic. Based on existing ICD datasets, this paper constructs a new dataset by additionally adding 100, 000 and 24, 252 hard negative pairs into the training and test set, respectively. Moreover, this paper further reveals a unique difficulty for solving the hard negative problem in ICD, i.e., there is a fundamental conflict between current metric learning and ICD. This conflict is: the metric learning adopts symmetric distance while the edited copy is an asymmetric (unidirectional) process, e.g., a partial crop is close to its holistic reference image and is an edited copy, while the latter cannot be the edited copy of the former (in spite the distance is equally small). This insight results in an Asymmetrical-Similarity Learning (ASL) method, which allows the similarity in two directions (the query <-> the reference image) to be different from each other. Experimental results show that ASL outperforms state-of-the-art methods by a clear margin, confirming that solving the symmetric-asymmetric conflict is critical for ICD. The NDEC dataset and code are available at //github.com/WangWenhao0716/ASL.
Many applications, such as system identification, classification of time series, direct and inverse problems in partial differential equations, and uncertainty quantification lead to the question of approximation of a non-linear operator between metric spaces $\mathfrak{X}$ and $\mathfrak{Y}$. We study the problem of determining the degree of approximation of such operators on a compact subset $K_\mathfrak{X}\subset \mathfrak{X}$ using a finite amount of information. If $\mathcal{F}: K_\mathfrak{X}\to K_\mathfrak{Y}$, a well established strategy to approximate $\mathcal{F}(F)$ for some $F\in K_\mathfrak{X}$ is to encode $F$ (respectively, $\mathcal{F}(F)$) in terms of a finite number $d$ (repectively $m$) of real numbers. Together with appropriate reconstruction algorithms (decoders), the problem reduces to the approximation of $m$ functions on a compact subset of a high dimensional Euclidean space $\mathbb{R}^d$, equivalently, the unit sphere $\mathbb{S}^d$ embedded in $\mathbb{R}^{d+1}$. The problem is challenging because $d$, $m$, as well as the complexity of the approximation on $\mathbb{S}^d$ are all large, and it is necessary to estimate the accuracy keeping track of the inter-dependence of all the approximations involved. In this paper, we establish constructive methods to do this efficiently; i.e., with the constants involved in the estimates on the approximation on $\mathbb{S}^d$ being $\mathcal{O}(d^{1/6})$. We study different smoothness classes for the operators, and also propose a method for approximation of $\mathcal{F}(F)$ using only information in a small neighborhood of $F$, resulting in an effective reduction in the number of parameters involved.
This article develops a new algorithm named TTRISK to solve high-dimensional risk-averse optimization problems governed by differential equations (ODEs and/or PDEs) under uncertainty. As an example, we focus on the so-called Conditional Value at Risk (CVaR), but the approach is equally applicable to other coherent risk measures. Both the full and reduced space formulations are considered. The algorithm is based on low rank tensor approximations of random fields discretized using stochastic collocation. To avoid non-smoothness of the objective function underpinning the CVaR, we propose an adaptive strategy to select the width parameter of the smoothed CVaR to balance the smoothing and tensor approximation errors. Moreover, unbiased Monte Carlo CVaR estimate can be computed by using the smoothed CVaR as a control variate. To accelerate the computations, we introduce an efficient preconditioner for the KKT system in the full space formulation.The numerical experiments demonstrate that the proposed method enables accurate CVaR optimization constrained by large-scale discretized systems. In particular, the first example consists of an elliptic PDE with random coefficients as constraints. The second example is motivated by a realistic application to devise a lockdown plan for United Kingdom under COVID-19. The results indicate that the risk-averse framework is feasible with the tensor approximations under tens of random variables.
Micron-scale robots (ubots) have recently shown great promise for emerging medical applications, and accurate control of ubots is a critical next step to deploying them in real systems. In this work, we develop the idea of a nonlinear mismatch controller to compensate for the mismatch between the disturbed unicycle model of a rolling ubot and trajectory data collected during an experiment. We exploit the differential flatness property of the rolling ubot model to generate a mapping from the desired state trajectory to nominal control actions. Due to model mismatch and parameter estimation error, the nominal control actions will not exactly reproduce the desired state trajectory. We employ a Gaussian Process (GP) to learn the model mismatch as a function of the desired control actions, and correct the nominal control actions using a least-squares optimization. We demonstrate the performance of our online learning algorithm in simulation, where we show that the model mismatch makes some desired states unreachable. Finally, we validate our approach in an experiment and show that the error metrics are reduced by up to 40%.
Approximate Bayesian Computation (ABC) enables statistical inference in simulator-based models whose likelihoods are difficult to calculate but easy to simulate from. ABC constructs a kernel-type approximation to the posterior distribution through an accept/reject mechanism which compares summary statistics of real and simulated data. To obviate the need for summary statistics, we directly compare empirical distributions with a Kullback-Leibler (KL) divergence estimator obtained via contrastive learning. In particular, we blend flexible machine learning classifiers within ABC to automate fake/real data comparisons. We consider the traditional accept/reject kernel as well as an exponential weighting scheme which does not require the ABC acceptance threshold. Our theoretical results show that the rate at which our ABC posterior distributions concentrate around the true parameter depends on the estimation error of the classifier. We derive limiting posterior shape results and find that, with a properly scaled exponential kernel, asymptotic normality holds. We demonstrate the usefulness of our approach on simulated examples as well as real data in the context of stock volatility estimation.
Mixed-Integer Linear Programming (MILP) plays an important role across a range of scientific disciplines and within areas of strategic importance to society. The MILP problems, however, suffer from combinatorial complexity. Because of integer decision variables, as the problem size increases, the number of possible solutions increases super-linearly thereby leading to a drastic increase in the computational effort. To efficiently solve MILP problems, a "price-based" decomposition and coordination approach is developed to exploit 1. the super-linear reduction of complexity upon the decomposition and 2. the geometric convergence potential inherent to Polyak's stepsizing formula for the fastest coordination possible to obtain near-optimal solutions in a computationally efficient manner. Unlike all previous methods to set stepsizes heuristically by adjusting hyperparameters, the key novel way to obtain stepsizes is purely decision-based: a novel "auxiliary" constraint satisfaction problem is solved, from which the appropriate stepsizes are inferred. Testing results for large-scale Generalized Assignment Problems (GAP) demonstrate that for the majority of instances, certifiably optimal solutions are obtained. For stochastic job-shop scheduling as well as for pharmaceutical scheduling, computational results demonstrate the two orders of magnitude speedup as compared to Branch-and-Cut (B&C). The new method has a major impact on the efficient resolution of complex Mixed-Integer Programming (MIP) problems arising within a variety of scientific fields.
With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191