We present a computationally efficient, data-driven procedure for constructing linear isometric embeddings of high-dimensional data (data frames) into spaces of smooth images, and thereby obtain tight frame dictionaries for the data space using tight frame dictionaries for the image space (“framed frames” – wavelets, curvelets, shearlets, etc.). Experiments indicate that data are more compressible in these induced dictionaries when compared to compressibility in terms of principal components.
We present an efficient algorithm and theory for Geometric Multi-Resolution Analysis (GMRA), a procedure for dictionary learning. Sparse dictionary learning provides the necessary complexity reduction for the critical applications of compression, regression, and classification in high-dimensional data analysis. As such, it is a critical technique in data science and it is important to have techniques that admit both efficient implementation and strong theory for large classes of theoretical models. By construction, GMRA is computationally efficient and in this paper we describe how the GMRA correctly approximates a large class of plausible models (namely, the noisy manifolds).
We introduce a procedure for learning discrete convolutional operators for generic datasets which recovers the standard block convolutional operators when applied to sets of natural images. They key observation is that the standard block convolutional operators on images are intuitive because humans naturally understand the grid structure of the self-evident functions over images spaces (pixels). This procedure first constructs a Geometric Multi-Resolution Analysis (GMRA) on the set of variables giving rise to a dataset, and then leverages the details of this data structure to identify subsets of variables upon which convolutional operators are supported, as well as a space of functions that can be shared coherently amongst these supports.
We show that the spaces of finite unit norm tight frames are connected, which verifies a conjecture first appearing in Dykema and Strawn (2006). Our central technique involves continuous liftings of paths from the polytope of eigensteps (see Cahill et al., 2012), or Gelfand-Tsetlin patterns, to spaces of FUNTFs. After demonstrating this connectivity result, we refine our analysis to show that the set of nonsingular points on these spaces is also connected, and we use this result to show that spaces of FUNTFs are irreducible in the algebro-geometric sense.
This paper introduces a method to integrate target behavior into the multiple hypothesis tracker (MHT) likelihood ratio. In particular, a periodic track appraisal based on behavior is introduced that uses elementary topological data analysis coupled with basic machine learning techniques. The track appraisal adjusts the traditional kinematic data association likelihood (i.e., track score) using an established formulation for classification-aided data association. The proposed method is tested and demonstrated on synthetic vehicular data representing an urban traffic scene generated by the Simulation of Urban Mobility package. The vehicles in the scene exhibit different driving behaviors. The proposed method distinguishes those behaviors and shows improved data association decisions relative to a conventional, kinematic MHT.
A finite (μ; S)-frame variety consists of the real or complex matrices F = [f1...fN] with frame operator FF* =
S, and satisfying IIfiII = μi for all i = 1,...,N. Here, S is a fixed Hermitian positive definite matrix and
μ = [μ1,..., μN] is a fixed list of lengths. These spaces generalize the well-known spaces of finite unit norm tight
frames. We explore the local geometry of these spaces and develop geometric optimization algorithms based on
the resulting insights.