arXiv subjects

Henry Kvinge

Publications and source records attributed to Henry Kvinge.

62 records · Page 4Linked to original sources

arXiv2018

Monitoring the shape of weather, soundscapes, and dynamical systems: a new statistic for dimension-driven data analysis on large data sets

Dimensionality-reduction methods are a fundamental tool in the analysis of large data sets. These algorithms work on the assumption that the "intrinsic dimension" of the data is generally much smaller than the ambient dimension in which it is collected. Alongside their usual purpose of mapping data into a smaller dimension with minimal information loss, dimensionality-reduction techniques implicitly or explicitly provide information about the dimension of the data set. In this paper, we propose a new statistic that we call the $κ$-profile for analysis of large data sets. The $κ$-profile arises from a dimensionality-reduction optimization problem: namely that of finding a projection into $k$-dimensions that optimally preserves the secants between points in the data set. From this optimal projection we extract $κ,$ the norm of the shortest projected secant from among the set of all normalized secants. This $κ$ can be computed for any $k$; thus the tuple of $κ$ values (indexed by dimension) becomes a $κ$-profile. Algorithms such as the Secant-Avoidance Projection algorithm and the Hierarchical Secant-Avoidance Projection algorithm, provide a computationally feasible means of estimating the $κ$-profile for large data sets, and thus a method of understanding and monitoring their behavior. As we demonstrate in this paper, the $κ$-profile serves as a useful statistic in several representative settings: weather data, soundscape data, and dynamical systems data.

cs.LG↗

arXiv2018

Coherent systems of probability measures on graphs for representations of free Frobenius towers

First formally defined by Borodin and Olshanski, a coherent system on a graded graph is a sequence of probability measures which respect the action of certain down/up transition functions between graded components. In one common example of such a construction, each measure is the Plancherel measure for the symmetric group $S_{n}$ and the down transition function is induced from the inclusions $S_{n} \hookrightarrow S_{n+1}$. In this paper we generalize the above framework to the case where $\{A_n\}_{n \geq 0}$ is any free Frobenius tower and $A_n$ is no longer assumed to be semisimple. In particular, we describe two coherent systems on graded graphs defined by the representation theory of $\{A_n\}_{n \geq 0}$ and connect one of these systems to a family of central elements of $\{A_n\}_{n \geq 0}$. When the algebras $\{A_n\}_{n \geq 0}$ are not semisimple, the resulting coherent systems reflect the duality between simple $A_n$-modules and indecomposable projective $A_n$-modules.

math.RT↗

arXiv2018

Too many secants: a hierarchical approach to secant-based dimensionality reduction on large data sets

A fundamental question in many data analysis settings is the problem of discerning the "natural" dimension of a data set. That is, when a data set is drawn from a manifold (possibly with noise), a meaningful aspect of the data is the dimension of that manifold. Various approaches exist for estimating this dimension, such as the method of Secant-Avoidance Projection (SAP). Intuitively, the SAP algorithm seeks to determine a projection which best preserves the lengths of all secants between points in a data set; by applying the algorithm to find the best projections to vector spaces of various dimensions, one may infer the dimension of the manifold of origination. That is, one may learn the dimension at which it is possible to construct a diffeomorphic copy of the data in a lower-dimensional Euclidean space. Using Whitney's embedding theorem, we can relate this information to the natural dimension of the data. A drawback of the SAP algorithm is that a data set with $T$ points has $O(T^2)$ secants, making the computation and storage of all secants infeasible for very large data sets. In this paper, we propose a novel algorithm that generalizes the SAP algorithm with an emphasis on addressing this issue. That is, we propose a hierarchical secant-based dimensionality-reduction method, which can be employed for data sets where explicitly calculating all secants is not feasible.

cs.CV↗

arXiv2018

The center of the twisted Heisenberg category, factorial Schur $Q$-functions, and transition functions on the Schur graph

We establish an isomorphism between the center of the twisted Heisenberg category and the subalgebra of the symmetric functions $Γ$ generated by odd power sums. We give a graphical description of the factorial Schur $Q$-functions as closed diagrams in the twisted Heisenberg category and show that the bubble generators of the center correspond to two sets of generators of $Γ$ which encode data related to up/down transition functions on the Schur graph. Finally, we describe an action of the trace of the twisted Heisenberg category, the $W$-algebra $W^-\subset W_{1+\infty}$, on $Γ$.

math.RT↗

arXiv2018

A GPU-Oriented Algorithm Design for Secant-Based Dimensionality Reduction

Dimensionality-reduction techniques are a fundamental tool for extracting useful information from high-dimensional data sets. Because secant sets encode manifold geometry, they are a useful tool for designing meaningful data-reduction algorithms. In one such approach, the goal is to construct a projection that maximally avoids secant directions and hence ensures that distinct data points are not mapped too close together in the reduced space. This type of algorithm is based on a mathematical framework inspired by the constructive proof of Whitney's embedding theorem from differential topology. Computing all (unit) secants for a set of points is by nature computationally expensive, thus opening the door for exploitation of GPU architecture for achieving fast versions of these algorithms. We present a polynomial-time data-reduction algorithm that produces a meaningful low-dimensional representation of a data set by iteratively constructing improved projections within the framework described above. Key to our algorithm design and implementation is the use of GPUs which, among other things, minimizes the computational time required for the calculation of all secant lines. One goal of this report is to share ideas with GPU experts and to discuss a class of mathematical algorithms that may be of interest to the broader GPU community.

cs.CV↗

arXiv2018

Endmember Extraction on the Grassmannian

Endmember extraction plays a prominent role in a variety of data analysis problems as endmembers often correspond to data representing the purest or best representative of some feature. Identifying endmembers then can be useful for further identification and classification tasks. In settings with high-dimensional data, such as hyperspectral imagery, it can be useful to consider endmembers that are subspaces as they are capable of capturing a wider range of variations of a signature. The endmember extraction problem in this setting thus translates to finding the vertices of the convex hull of a set of points on a Grassmannian. In the presence of noise, it can be less clear whether a point should be considered a vertex. In this paper, we propose an algorithm to extract endmembers on a Grassmannian, identify subspaces of interest that lie near the boundary of a convex hull, and demonstrate the use of the algorithm on a synthetic example and on the 220 spectral band AVIRIS Indian Pines hyperspectral image.

cs.CV↗

arXiv2016

Khovanov's Heisenberg category, moments in free probability, and shifted symmetric functions

We establish an isomorphism between the center of the Heisenberg category defined by Khovanov and the algebra $Λ^*$ of shifted symmetric functions defined by Okounkov-Olshanski. We give a graphical description of the shifted power and Schur bases of $Λ^*$ as elements of the center, and describe the curl generators of the center in the language of shifted symmetric functions. This latter description makes use of the transition and co-transition measures of Kerov and the noncommutative probability spaces of Biane.

math.RT↗

arXiv2015

Categorifying the tensor product of the Kirillov-Reshetikhin crystal $B^{1,1}$ and a fundamental crystal

We use Khovanov-Lauda-Rouquier (KLR) algebras to categorify a crystal isomorphism between a fundamental crystal and the tensor product of a Kirillov-Reshetikhin crystal and another fundamental crystal, all in affine type. The nodes of the Kirillov-Reshetikhin crystal correspond to a family of "trivial" modules. The nodes of the fundamental crystal correspond to simple modules of the corresponding cyclotomic KLR algebra. The crystal operators correspond to socle of restriction and behave compatibly with the rule for tensor product of crystal graphs.

math.RT↗