Refined Complexity of PCA with Outliers

Principal component analysis (PCA) is one of the most fundamental procedures in exploratory data analysis and is the basic step in applications ranging from quantitative finance and bioinformatics to image analysis and neuroscience. However, it is well-documented that the applicability of PCA in many real scenarios could be constrained by an "immune deficiency" to outliers such as corrupted observations. We consider the following algorithmic question about the PCA with outliers. For a set of $n$ points in $\mathbb{R}^{d}$, how to learn a subset of points, say 1% of the total number of points, such that the remaining part of the points is best fit into some unknown $r$-dimensional subspace? We provide a rigorous algorithmic analysis of the problem. We show that the problem is solvable in time $n^{O(d^2)}$. In particular, for constant dimension the problem is solvable in polynomial time. We complement the algorithmic result by the lower bound, showing that unless Exponential Time Hypothesis fails, in time $f(d)n^{o(d)}$, for any function $f$ of $d$, it is impossible not only to solve the problem exactly but even to approximate it within a constant factor.Comment: To be presented at ICML 201

Complexity of Token Swapping and its Variants

In the Token Swapping problem we are given a graph with a token placed on each vertex. Each token has exactly one destination vertex, and we try to move all the tokens to their destinations, using the minimum number of swaps, i.e., operations of exchanging the tokens on two adjacent vertices. As the main result of this paper, we show that Token Swapping is $W[1]$-hard parameterized by the length $k$ of a shortest sequence of swaps. In fact, we prove that, for any computable function $f$, it cannot be solved in time $f(k)n^{o(k / \log k)}$ where $n$ is the number of vertices of the input graph, unless the ETH fails. This lower bound almost matches the trivial $n^{O(k)}$-time algorithm. We also consider two generalizations of the Token Swapping, namely Colored Token Swapping (where the tokens have different colors and tokens of the same color are indistinguishable), and Subset Token Swapping (where each token has a set of possible destinations). To complement the hardness result, we prove that even the most general variant, Subset Token Swapping, is FPT in nowhere-dense graph classes. Finally, we consider the complexities of all three problems in very restricted classes of graphs: graphs of bounded treewidth and diameter, stars, cliques, and paths, trying to identify the borderlines between polynomial and NP-hard cases.Comment: 23 pages, 7 Figure

East African topography and volcanism explained by a single, migrating plume

Anomalous topographic swells and Cenozoic volcanism in east Africa have been associated with mantle plumes. Several models involving one or more fixed plumes beneath the northeastward migrating African plate have been suggested to explain the space-time distribution of magmatism in east Africa. We devise paleogeographically constrained global models of mantle convection and, based on the evolution of flow in the deepest lower mantle, show that the Afar plume migrated southward throughout its lifetime. The models suggest that the mobile Afar plume provides a dynamically consistent explanation for the spatial extent of the southward propagation of the east African rift system (EARS), which is difficult to explain by the northeastward migration of Africa over one or more fixed plumes alone, over the last ≈45 Myrs. We further show that the age-progression of volcanism associated with the southward propagation of EARS is consistent with the apparent surface hotspot motion that results from southward motion of the modelled Afar plume beneath the northeastward migrating African plate. The models suggest that the Afar plume became weaker as it migrated southwards, consistent with trends observed in the geochemical record

Axiomatic Digital Topology

The paper presents a new set of axioms of digital topology, which are easily understandable for application developers. They define a class of locally finite (LF) topological spaces. An important property of LF spaces satisfying the axioms is that the neighborhood relation is antisymmetric and transitive. Therefore any connected and non-trivial LF space is isomorphic to an abstract cell complex. The paper demonstrates that in an n-dimensional digital space only those of the (a, b)-adjacencies commonly used in computer imagery have analogs among the LF spaces, in which a and b are different and one of the adjacencies is the "maximal" one, corresponding to 3n\"i1 neighbors. Even these (a, b)-adjacencies have important limitations and drawbacks. The most important one is that they are applicable only to binary images. The way of easily using LF spaces in computer imagery on standard orthogonal grids containing only pixels or voxels and no cells of lower dimensions is suggested

Metallic nanoparticles meet Metadynamics

We show how standard Metadynamics coupled with classical Molecular Dynamics can be successfully ap- plied to sample the configurational and free energy space of metallic and bimetallic nanopclusters via the implementation of collective variables related to the pair distance distribution function of the nanoparticle itself. As paradigmatic examples we show an application of our methodology to Ag147, Pt147 and their alloy AgshellPtcore at 1:1 and 2:1 chemical compositions. The proposed scheme is not only able to reproduce known structural transformation pathways, as the five and the six square-diamond mechanisms both in pure and core-shell nanoparticles but also to predict a new route connecting icosahedron to anti-cuboctahedron.Comment: 7 pages, 8 figure

Double Scaling in Tensor Models with a Quartic Interaction

In this paper we identify and analyze in detail the subleading contributions in the 1/N expansion of random tensors, in the simple case of a quartically interacting model. The leading order for this 1/N expansion is made of graphs, called melons, which are dual to particular triangulations of the D-dimensional sphere, closely related to the "stacked" triangulations. For D<6 the subleading behavior is governed by a larger family of graphs, hereafter called cherry trees, which are also dual to the D-dimensional sphere. They can be resummed explicitly through a double scaling limit. In sharp contrast with random matrix models, this double scaling limit is stable. Apart from its unexpected upper critical dimension 6, it displays a singularity at fixed distance from the origin and is clearly the first step in a richer set of yet to be discovered multi-scaling limits

The double scaling limit of random tensor models

Tensor models generalize matrix models and generate colored triangulations of pseudo-manifolds in dimensions $D\geq 3$. The free energies of some models have been recently shown to admit a double scaling limit, i.e. large tensor size $N$ while tuning to criticality, which turns out to be summable in dimension less than six. This double scaling limit is here extended to arbitrary models. This is done by means of the Schwinger--Dyson equations, which generalize the loop equations of random matrix models, coupled to a double scale analysis of the cumulants.Comment: 37 pages, 13 figures; several references were added. A new subsection was added to first present all the results (before the technical proofs which will follow). A misprint was correcte