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Diffusion maps are an emerging data-driven technique for non-linear dimensionality reduction, which are especially useful for the analysis of coherent structures and nonlinear embeddings of dynamical systems. However, the computational…
High-dimensional reduction methods are powerful tools for describing the main patterns in big data. One of these methods is the topological data analysis (TDA), which modeling the shape of the data in terms of topological properties. This…
Neighbor embedding is widely employed to visualize high-dimensional data; however, it frequently overlooks the global structure, e.g., intercluster similarities, thereby impeding accurate visualization. To address this problem, this paper…
Subspace identification is a classical and very well studied problem in system identification. The problem was recently posed as a convex optimization problem via the nuclear norm relaxation. Inspired by robust PCA, we extend this framework…
We consider the problem of finding anomalies in high-dimensional data using popular PCA based anomaly scores. The naive algorithms for computing these scores explicitly compute the PCA of the covariance matrix which uses space quadratic in…
With the recent surge in big data analytics for hyper-dimensional data there is a renewed interest in dimensionality reduction techniques for machine learning applications. In order for these methods to improve performance gains and…
Structured prediction tasks pose a fundamental trade-off between the need for model complexity to increase predictive power and the limited computational resources for inference in the exponentially-sized output spaces such models require.…
This paper provides a new way of developing the fast iterative shrinkage/thresholding algorithm (FISTA) that is widely used for minimizing composite convex functions with a nonsmooth term such as the $\ell_1$ regularizer. In particular,…
Image recognition systems have demonstrated tremendous progress over the past few decades thanks, in part, to our ability of learning compact and robust representations of images. As we witness the wide spread adoption of these systems, it…
It has been shown that dimension reduction methods such as PCA may be inherently prone to unfairness and treat data from different sensitive groups such as race, color, sex, etc., unfairly. In pursuit of fairness-enhancing dimensionality…
There is an increasing body of work exploring the integration of random projection into algorithms for numerical linear algebra. The primary motivation is to reduce the overall computational cost of processing large datasets. A suitably…
Iterative phase retrieval algorithms typically employ projections onto constraint subspaces to recover the unknown phases in the Fourier transform of an image, or, in the case of x-ray crystallography, the electron density of a molecule.…
One of the challenges in analyzing high-dimensional expression data is the detection of important biological signals. A common approach is to apply a dimension reduction method, such as principal component analysis. Typically, after…
Spatial Transcriptomics (ST) is a method that captures gene expression profiles aligned with spatial coordinates. The discrete spatial distribution and the super-high dimensional sequencing results make ST data challenging to be modeled…
In recent years, large high-dimensional data sets have become commonplace in a wide range of applications in science and commerce. Techniques for dimension reduction are of primary concern in statistical analysis. Projection methods play an…
Principal Component Analysis (PCA) is a widely utilized technique for dimensionality reduction; however, its inherent lack of interpretability-stemming from dense linear combinations of all feature-limits its applicability in many domains.…
Data visualization is the process by which data of any size or dimensionality is processed to produce an understandable set of data in a lower dimensionality, allowing it to be manipulated and understood more easily by people. The goal of…
The success of algorithms in the analysis of high-dimensional data is often attributed to the manifold hypothesis, which supposes that this data lie on or near a manifold of much lower dimension. It is often useful to determine or estimate…
Principal Component Analysis (PCA) is known to be the most widely applied dimensionality reduction approach. A lot of improvements have been done on the traditional PCA, in order to obtain optimal results in the dimensionality reduction of…
Dimensionality Reduction (DR) methods are widely used to visualize high-dimensional data. One key task in DR-based analysis is discovering neighborhoods, which relies on analyzing the fine-grained local structure of a projection. However,…