Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs

Laplacian mixture models identify overlapping regions of influence in unlabeled graph and network data in a scalable and computationally efficient way, yielding useful low-dimensional representations. By combining Laplacian eigenspace and finite mixture modeling methods, they provide probabilistic or fuzzy dimensionality reductions or domain decompositions for a variety of input data types, including mixture distributions, feature vectors, and graphs or networks. Provable optimal recovery using the algorithm is analytically shown for a nontrivial class of cluster graphs. Heuristic approximations for scalable high-performance implementations are described and empirically tested. Connections to PageRank and community detection in network analysis demonstrate the wide applicability of this approach. The origins of fuzzy spectral methods, beginning with generalized heat or diffusion equations in physics, are reviewed and summarized. Comparisons to other dimensionality reduction and clustering methods for challenging unsupervised machine learning problems are also discussed.Comment: 13 figures, 35 reference

Pointer basis induced by collisional decoherence

We study the emergence and dynamics of pointer states in the motion of a quantum test particle affected by collisional decoherence. These environmentally distinguished states are shown to be exponentially localized solitonic wave functions which evolve according to the classical equations of motion. We explain their formation using the orthogonal unraveling of the master equation, and we demonstrate that the statistical weights of the arising mixture are given by projections of the initial state onto the pointer basis.Comment: 26 pages, 9 figures; v2: minimal corrections, corresponds to published versio

Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State

A recent conjecture regarding the average of the minimum eigenvalue of the reduced density matrix of a random complex state is proved. In fact, the full distribution of the minimum eigenvalue is derived exactly for both the cases of a random real and a random complex state. Our results are relevant to the entanglement properties of eigenvectors of the orthogonal and unitary ensembles of random matrix theory and quantum chaotic systems. They also provide a rare exactly solvable case for the distribution of the minimum of a set of N {\em strongly correlated} random variables for all values of N (and not just for large N).Comment: 13 pages, 2 figures included; typos corrected; to appear in J. Stat. Phy

Pseudo-factorials, elliptic functions, and continued fractions

This study presents miscellaneous properties of pseudo-factorials, which are numbers whose recurrence relation is a twisted form of that of usual factorials. These numbers are associated with special elliptic functions, most notably, a Dixonian and a Weierstrass function, which parametrize the Fermat cubic curve and are relative to a hexagonal lattice. A continued fraction expansion of the ordinary generating function of pseudo-factorials, first discovered empirically, is established here. This article also provides a characterization of the associated orthogonal polynomials, which appear to form a new family of "elliptic polynomials", as well as various other properties of pseudo-factorials, including a hexagonal lattice sum expression and elementary congruences.Comment: 24 pages; with correction of typos and minor revision. To appear in The Ramanujan Journa

Slingshot: cell lineage and pseudotime inference for single-cell transcriptomics.

BackgroundSingle-cell transcriptomics allows researchers to investigate complex communities of heterogeneous cells. It can be applied to stem cells and their descendants in order to chart the progression from multipotent progenitors to fully differentiated cells. While a variety of statistical and computational methods have been proposed for inferring cell lineages, the problem of accurately characterizing multiple branching lineages remains difficult to solve.ResultsWe introduce Slingshot, a novel method for inferring cell lineages and pseudotimes from single-cell gene expression data. In previously published datasets, Slingshot correctly identifies the biological signal for one to three branching trajectories. Additionally, our simulation study shows that Slingshot infers more accurate pseudotimes than other leading methods.ConclusionsSlingshot is a uniquely robust and flexible tool which combines the highly stable techniques necessary for noisy single-cell data with the ability to identify multiple trajectories. Accurate lineage inference is a critical step in the identification of dynamic temporal gene expression

A gauge model for quantum mechanics on a stratified space

In the Hamiltonian approach on a single spatial plaquette, we construct a quantum (lattice) gauge theory which incorporates the classical singularities. The reduced phase space is a stratified K\"ahler space, and we make explicit the requisite singular holomorphic quantization procedure on this space. On the quantum level, this procedure furnishes a costratified Hilbert space, that is, a Hilbert space together with a system which consists of the subspaces associated with the strata of the reduced phase space and of the corresponding orthoprojectors. The costratified Hilbert space structure reflects the stratification of the reduced phase space. For the special case where the structure group is $\mathrm{SU}(2)$, we discuss the tunneling probabilities between the strata, determine the energy eigenstates and study the corresponding expectation values of the orthoprojectors onto the subspaces associated with the strata in the strong and weak coupling approximations.Comment: 38 pages, 9 figures. Changes: comments on the heat kernel and coherent states have been adde