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Approximate Bayesian computation (ABC) and other likelihood-free inference methods have gained popularity in the last decade, as they allow rigorous statistical inference for complex models without analytically tractable likelihood…
Bayesian inverse problems use observed data to update a prior probability distribution for an unknown state or parameter of a scientific system to a posterior distribution conditioned on the data. In many applications, the unknown parameter…
In this paper we study a new approach in optimization that aims to search a large domain D where a given function takes large, small or specific values via an iterative optimization algorithm based on the gradient. We show that the…
The rapidly evolving field of engineering design of functional surfaces necessitates sophisticated tools to manage the inherent complexity of high-dimensional design spaces. This survey paper offers a scoping review, i.e., a literature…
Classical dimensional analysis has two limitations: (i) the computed dimensionless groups are not unique, and (ii) the analysis does not measure relative importance of the dimensionless groups. We propose two algorithms for estimating…
Privacy preservation has become a critical concern in high-dimensional data analysis due to the growing prevalence of data-driven applications. Since its proposal, sliced inverse regression has emerged as a widely utilized statistical…
A change of the prevalent supervised learning techniques is foreseeable in the near future: from the complex, computational expensive algorithms to more flexible and elementary training ones. The strong revitalization of randomized…
Modeling data as being sampled from a union of independent subspaces has been widely applied to a number of real world applications. However, dimensionality reduction approaches that theoretically preserve this independence assumption have…
In this paper, we study randomized reduction methods, which reduce high-dimensional features into low-dimensional space by randomized methods (e.g., random projection, random hashing), for large-scale high-dimensional classification.…
Randomized dimensionality reduction is a widely-used algorithmic technique for speeding up large-scale Euclidean optimization problems. In this paper, we study dimension reduction for a variety of maximization problems, including…
It is often possible to perform reduced order modelling by specifying linear subspace which accurately captures the dynamics of the system. This approach becomes especially appealing when linear subspace explicitly depends on parameters of…
Parameter inference for dynamical models of (bio)physical systems remains a challenging problem. Intractable gradients, high-dimensional spaces, and non-linear model functions are typically problematic without large computational budgets. A…
High-dimensional big data appears in many research fields such as image recognition, biology and collaborative filtering. Often, the exploration of such data by classic algorithms is encountered with difficulties due to `curse of…
We introduce a method to construct a stochastic surrogate model from the results of dimensionality reduction in forward uncertainty quantification. The hypothesis is that the high-dimensional input augmented by the output of a computational…
In this paper (part 1), we describe a derivative-free trust-region method for solving unconstrained optimization problems. We will discuss a method when we relax the model order assumption and use artificial neural network techniques to…
This paper, broadly speaking, covers the use of randomness in two main areas: low-rank approximation and kernel methods. Low-rank approximation is very important in numerical linear algebra. Many applications depend on matrix decomposition…
In this note, we develop fast and deterministic dimensionality reduction techniques for a family of subspace approximation problems. Let $P\subset \mathbbm{R}^N$ be a given set of $M$ points. The techniques developed herein find an $O(n…
We study a natural extension of classical empirical risk minimization, where the hypothesis space is a random subspace of a given space. In particular, we consider possibly data dependent subspaces spanned by a random subset of the data,…
We study the following basic machine learning task: Given a fixed set of $d$-dimensional input points for a linear regression problem, we wish to predict a hidden response value for each of the points. We can only afford to attain the…
In this paper, we propose a novel lower dimensional representation of a shape sequence. The proposed dimension reduction is invertible and computationally more efficient in comparison to other related works. Theoretically, the differential…