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As distributed systems increase in size, the need for scalable algorithms becomes more and more important. We argue that in the context of system identification, an essential building block of any scalable algorithm is the ability to…
Sufficient dimension reduction aims for reduction of dimensionality of a regression without loss of information by replacing the original predictor with its lower-dimensional subspace. Partial (sufficient) dimension reduction arises when…
Low discrepancy point sets have been widely used as a tool to approximate continuous objects by discrete ones in numerical processes, for example in numerical integration. Following a century of research on the topic, it is still unclear…
Dimensionality reduction is an essential technique for multi-way large-scale data, i.e., tensor. Tensor ring (TR) decomposition has become popular due to its high representation ability and flexibility. However, the traditional TR…
A complex-valued signal is improper if it is correlated with its complex conjugate. The dimension of the improper signal subspace, i.e., the number of improper components in a complex-valued measurement, is an important parameter and is…
Neural networks achieve remarkable performance through superposition: encoding multiple features as overlapping directions in activation space rather than dedicating individual neurons to each feature. This challenges interpretability, yet…
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…
Superposition -- when a neural network represents more ``features'' than it has dimensions -- seems to pose a serious challenge to mechanistically interpreting current AI systems. Existing theory work studies \emph{representational}…
This article explores the generalized analysis-of-variance or ANOVA dimensional decomposition (ADD) for multivariate functions of dependent random variables. Two notable properties, stemming from weakened annihilating conditions, reveal…
Sufficient dimension reduction reduces the dimensionality of data while preserving relevant regression information. In this article, we develop Minimum Average Deviance Estimation (MADE) methodology for sufficient dimension reduction. It…
Multidimensional scaling (MDS) is a popular dimensionality reduction techniques that has been widely used for network visualization and cooperative localization. However, the traditional stress minimization formulation of MDS necessitates…
Recent studies have highlighted the benefits of generating multiple synthetic datasets for supervised learning, from increased accuracy to more effective model selection and uncertainty estimation. These benefits have clear empirical…
This paper presents a randomized algorithm for computing the near-optimal low-rank dynamic mode decomposition (DMD). Randomized algorithms are emerging techniques to compute low-rank matrix approximations at a fraction of the cost of…
The problem of decomposing non-manifold object has already been studied in solid modeling. However, the few proposed solutions are limited to the problem of decomposing solids described through their boundaries. In this thesis we study the…
We propose an algorithmic framework, that employs active subspace techniques, for scalable global optimization of functions with low effective dimension (also referred to as low-rank functions). This proposal replaces the original…
The active element pattern method is widely employed in beam pattern synthesis of array antenna to account for mutual coupling between antenna elements. Calculating the active element patterns for large number of array requires full-wave…
Tensor decomposition methods are popular tools for learning latent variables given only lower-order moments of the data. However, the standard assumption is that we have sufficient data to estimate these moments to high accuracy. In this…
A key step in reverse engineering neural networks is to decompose them into simpler parts that can be studied in relative isolation. Linear parameter decomposition -- a framework that has been proposed to resolve several issues with current…
We formulate a low-storage method for performing dynamic mode decomposition that can be updated inexpensively as new data become available; this formulation allows dynamical information to be extracted from large datasets and data streams.…
To estimate casual treatment effects, we propose a new matching approach based on the reduced covariates obtained from sufficient dimension reduction. Compared to the original covariates and the propensity score, which are commonly used for…