Classification and Geometry of General Perceptual Manifolds

Perceptual manifolds arise when a neural population responds to an ensemble of sensory signals associated with different physical features (e.g., orientation, pose, scale, location, and intensity) of the same perceptual object. Object recognition and discrimination requires classifying the manifolds in a manner that is insensitive to variability within a manifold. How neuronal systems give rise to invariant object classification and recognition is a fundamental problem in brain theory as well as in machine learning. Here we study the ability of a readout network to classify objects from their perceptual manifold representations. We develop a statistical mechanical theory for the linear classification of manifolds with arbitrary geometry revealing a remarkable relation to the mathematics of conic decomposition. Novel geometrical measures of manifold radius and manifold dimension are introduced which can explain the classification capacity for manifolds of various geometries. The general theory is demonstrated on a number of representative manifolds, including L2 ellipsoids prototypical of strictly convex manifolds, L1 balls representing polytopes consisting of finite sample points, and orientation manifolds which arise from neurons tuned to respond to a continuous angle variable, such as object orientation. The effects of label sparsity on the classification capacity of manifolds are elucidated, revealing a scaling relation between label sparsity and manifold radius. Theoretical predictions are corroborated by numerical simulations using recently developed algorithms to compute maximum margin solutions for manifold dichotomies. Our theory and its extensions provide a powerful and rich framework for applying statistical mechanics of linear classification to data arising from neuronal responses to object stimuli, as well as to artificial deep networks trained for object recognition tasks.Comment: 24 pages, 12 figures, Supplementary Material

An Information Maximization Approach to Overcomplete and Recurrent Representations

The principle of maximizing mutual information is applied to learning overcomplete and recurrent representations. The underlying model consists of a network of input units driving a larger number of output units with recurrent interactions. In the limit of zero noise, the network is deterministic and the mutual information can be related to the entropy of the output units. Maximizing this entropy with respect to both the feedforward connections as well as the recurrent interactions results in simple learning rules for both sets of parameters. The conventional independent components (ICA) learning algorithm can be recovered as a special case where there is an equal number of output units and no recurrent connections. The application of these new learning rules is illustrated on a simple two-dimensional input example

Short-Term Memory in Orthogonal Neural Networks

We study the ability of linear recurrent networks obeying discrete time dynamics to store long temporal sequences that are retrievable from the instantaneous state of the network. We calculate this temporal memory capacity for both distributed shift register and random orthogonal connectivity matrices. We show that the memory capacity of these networks scales with system size.Comment: 4 pages, 4 figures, to be published in Phys. Rev. Let

An empirical analysis of smart contracts: platforms, applications, and design patterns

Smart contracts are computer programs that can be consistently executed by a network of mutually distrusting nodes, without the arbitration of a trusted authority. Because of their resilience to tampering, smart contracts are appealing in many scenarios, especially in those which require transfers of money to respect certain agreed rules (like in financial services and in games). Over the last few years many platforms for smart contracts have been proposed, and some of them have been actually implemented and used. We study how the notion of smart contract is interpreted in some of these platforms. Focussing on the two most widespread ones, Bitcoin and Ethereum, we quantify the usage of smart contracts in relation to their application domain. We also analyse the most common programming patterns in Ethereum, where the source code of smart contracts is available.Comment: WTSC 201

Subextensive singularity in the 2D ±J\pm J Ising spin glass

The statistics of low energy states of the 2D Ising spin glass with +1 and -1 bonds are studied for $L \times L$ square lattices with $L \le 48$, and $p$ = 0.5, where $p$ is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. The behavior of the density of states near the ground state energy is analyzed as a function of $L$, in order to obtain the low temperature behavior of the model. For large finite $L$ there is a range of $T$ in which the heat capacity is proportional to $T^{5.33 \pm 0.12}$. The range of $T$ in which this behavior occurs scales slowly to $T = 0$ as $L$ increases. Similar results are found for $p$ = 0.25. Our results indicate that this model probably obeys the ordinary hyperscaling relation $d \nu = 2 - \alpha$, even though $T_c = 0$. The existence of the subextensive behavior is attributed to long-range correlations between zero-energy domain walls, and evidence of such correlations is presented.Comment: 13 pages, 7 figures; final version, to appear in J. Stat. Phy

A Logic of Blockchain Updates

Blockchains are distributed data structures that are used to achieve consensus in systems for cryptocurrencies (like Bitcoin) or smart contracts (like Ethereum). Although blockchains gained a lot of popularity recently, there is no logic-based model for blockchains available. We introduce BCL, a dynamic logic to reason about blockchain updates, and show that BCL is sound and complete with respect to a simple blockchain model

Decentralization in Bitcoin and Ethereum Networks

Blockchain-based cryptocurrencies have demonstrated how to securely implement traditionally centralized systems, such as currencies, in a decentralized fashion. However, there have been few measurement studies on the level of decentralization they achieve in practice. We present a measurement study on various decentralization metrics of two of the leading cryptocurrencies with the largest market capitalization and user base, Bitcoin and Ethereum. We investigate the extent of decentralization by measuring the network resources of nodes and the interconnection among them, the protocol requirements affecting the operation of nodes, and the robustness of the two systems against attacks. In particular, we adapted existing internet measurement techniques and used the Falcon Relay Network as a novel measurement tool to obtain our data. We discovered that neither Bitcoin nor Ethereum has strictly better properties than the other. We also provide concrete suggestions for improving both systems.Comment: Financial Cryptography and Data Security 201

On the distribution of barriers in the spin glasses

We discuss a general formalism that allows study of transitions over barriers in spin glasses with long-range interactions that contain large but finite number, $N$, of spins. We apply this formalism to the Sherrington-Kirkpatrick model with finite $N$ and derive equations for the dynamical order parameters which allow ''instanton'' solutions describing transitions over the barriers separating metastable states. Specifically, we study these equations for a glass state that was obtained in a slow cooling process ending a little below $T_{c}$ and show that these equations allow ''instanton'' solutions which erase the response of the glass to the perturbations applied during the slow cooling process. The corresponding action of these solutions gives the energy of the barriers, we find that it scales as $\tau ^{6}$ where $\tau$ is the reduced temperature.Comment: 8 pages, LaTex, 2 Postscript figure