17,122 research outputs found
Machine learning in space forms: Embeddings, classification, and similarity comparisons
We take a non-Euclidean view at three classical machine learning subjects: low-dimensional embedding, classification, and similarity comparisons.
We first introduce kinetic Euclidean distance matrices to solve kinetic distance geometry problems. In distance geometry problems (DGPs), the task is to find a geometric representation, that is, an embedding, for a collection of entities consistent with pairwise distance (metric) or similarity (nonmetric) measurements. In kinetic DGPs, the twist is that the points are dynamic. And our goal is to localize them by exploiting the information about their trajectory class. We show that a semidefinite relaxation can reconstruct trajectories from incomplete, noisy, time-varying distance observations. We then introduce another distance-geometric object: hyperbolic distance matrices. Recent works have focused on hyperbolic embedding methods for low-distortion embedding of distance measurements associated with hierarchical data. We derive a semidefinite relaxation to estimate the missing distance measurements and denoise them. Further, we formalize the hyperbolic Procrustes analysis, which uses extraneous information in the form of anchor points, to uniquely identify the embedded points.
Next, we address the design of learning algorithms in mixed-curvature spaces. Learning algorithms in low-dimensional mixed-curvature spaces have been limited to certain non-Euclidean neural networks. Here, we study the problem of learning a linear classifier (a perceptron) in product of Euclidean, spherical, and hyperbolic spaces, i.e., space forms. We introduce a notion of linear separation surfaces in Riemannian manifolds and use a metric that renders distances in different space forms compatible with each other and integrates them into one classifier.
Lastly, we show how similarity comparisons carry information about the underlying space of geometric graphs. We introduce the ordinal spread of a distance list and relate it to the ordinal capacity of their underlying space, a notion that quantifies the space's ability to host extreme patterns in nonmetric measurements. Then, we use the distribution of random ordinal spread variables as a practical tool to identify the underlying space form
Oscillating about coplanarity in the 4 body problem
For the Newtonian 4-body problem in space we prove that any zero angular
momentum bounded solution suffers infinitely many coplanar instants, that is,
times at which all 4 bodies lie in the same plane. This result generalizes a
known result for collinear instants ("syzygies") in the zero angular momentum
planar 3-body problem, and extends to the body problem in -space. The
proof, for , starts by identifying the center-of-mass zero configuration
space with real matrices, the coplanar configurations with
matrices whose determinant is zero, and the mass metric with the Frobenius
(standard Euclidean) norm. Let denote the signed distance from a matrix to
the hypersurface of matrices with determinant zero. The proof hinges on
establishing a harmonic oscillator type ODE for along solutions. Bounds on
inter-body distances then yield an explicit lower bound for the
frequency of this oscillator, guaranteeing a degeneration within every time
interval of length . The non-negativity of the curvature of
oriented shape space (the quotient of configuration space by the rotation
group) plays a crucial role in the proof.Comment: 26 pages, 5 figure
Quantization of systems with internal degrees of freedom in two-dimensional manifolds
Presented is a primary step towards quantization of infinitesimal rigid body
moving in a two-dimensional manifold. The special stress is laid on spaces of
constant curvature like the two-dimensional sphere and pseudosphere
(Lobatschevski space). Also two-dimensional torus is briefly discussed as an
interesting algebraic manifold.Comment: 19 page
Membrane Scattering in Curved Space with M-Momentum Transfer
We study membrane scattering in a curved space with non-zero M-momentum
p_{11} transfer. In the low-energy short-distance region, the membrane dynamics
is described by a three-dimensional N=4 supersymmetric gauge theory. We study
an n-instanton process of the gauge theory, corresponding to the exchange of n
units of p_{11}, and compare the result with the scattering amplitude computed
in the low-energy long-distance region using supergravity. We find that they
behave differently. We show that this result is not in contradiction with the
large-N Matrix Theory conjecture, by pointing out that cutoff scales of the
supergravity and the gauge theory are complementary to each other.Comment: 30 pages, 2 postscript figures, late
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