Minimal Algorithmic Information Loss Methods for Dimension Reduction, Feature Selection and Network Sparsification

We introduce a family of unsupervised, domain-free, and (asymptotically) model-independent algorithms based on the principles of algorithmic probability and information theory designed to minimize the loss of algorithmic information, including a lossless-compression-based lossy compression algorithm. The methods can select and coarse-grain data in an algorithmic-complexity fashion (without the use of popular compression algorithms) by collapsing regions that may procedurally be regenerated from a computable candidate model. We show that the method can preserve the salient properties of objects and perform dimension reduction, denoising, feature selection, and network sparsification. As validation case, we demonstrate that the method preserves all the graph-theoretic indices measured on a well-known set of synthetic and real-world networks of very different nature, ranging from degree distribution and clustering coefficient to edge betweenness and degree and eigenvector centralities, achieving equal or significantly better results than other data reduction and some of the leading network sparsification methods. The methods (InfoRank, MILS) can also be applied to applications such as image segmentation based on algorithmic probability.Comment: 23 pages in double column including Appendix, online implementation at http://complexitycalculator.com/MILS

Sparsity-Sensitive Finite Abstraction

Abstraction of a continuous-space model into a finite state and input dynamical model is a key step in formal controller synthesis tools. To date, these software tools have been limited to systems of modest size (typically $\leq$ 6 dimensions) because the abstraction procedure suffers from an exponential runtime with respect to the sum of state and input dimensions. We present a simple modification to the abstraction algorithm that dramatically reduces the computation time for systems exhibiting a sparse interconnection structure. This modified procedure recovers the same abstraction as the one computed by a brute force algorithm that disregards the sparsity. Examples highlight speed-ups from existing benchmarks in the literature, synthesis of a safety supervisory controller for a 12-dimensional and abstraction of a 51-dimensional vehicular traffic network

A hierarchy of topological tensor network states

We present a hierarchy of quantum many-body states among which many examples of topological order can be identified by construction. We define these states in terms of a general, basis-independent framework of tensor networks based on the algebraic setting of finite-dimensional Hopf C*-algebras. At the top of the hierarchy we identify ground states of new topological lattice models extending Kitaev's quantum double models [26]. For these states we exhibit the mechanism responsible for their non-zero topological entanglement entropy by constructing a renormalization group flow. Furthermore it is shown that those states of the hierarchy associated with Kitaev's original quantum double models are related to each other by the condensation of topological charges. We conjecture that charge condensation is the physical mechanism underlying the hierarchy in general.Comment: 61 page

2D Conformal Field Theories and Holography

It is known that the chiral part of any 2d conformal field theory defines a 3d topological quantum field theory: quantum states of this TQFT are the CFT conformal blocks. The main aim of this paper is to show that a similar CFT/TQFT relation exists also for the full CFT. The 3d topological theory that arises is a certain ``square'' of the chiral TQFT. Such topological theories were studied by Turaev and Viro; they are related to 3d gravity. We establish an operator/state correspondence in which operators in the chiral TQFT correspond to states in the Turaev-Viro theory. We use this correspondence to interpret CFT correlation functions as particular quantum states of the Turaev-Viro theory. We compute the components of these states in the basis in the Turaev-Viro Hilbert space given by colored 3-valent graphs. The formula we obtain is a generalization of the Verlinde formula. The later is obtained from our expression for a zero colored graph. Our results give an interesting ``holographic'' perspective on conformal field theories in 2 dimensions.Comment: 29+1 pages, many figure