Are single-peaked tuning curves tuned for speed rather than accuracy?

According to the efficient coding hypothesis, sensory neurons are adapted to provide maximal information about the environment, given some biophysical constraints. In early visual areas, stimulus-induced modulations of neural activity (or tunings) are predominantly single-peaked. However, periodic tuning, as exhibited by grid cells, has been linked to a significant increase in decoding performance. Does this imply that the tuning curves in early visual areas are sub-optimal? We argue that the time scale at which neurons encode information is imperative to understand the advantages of single-peaked and periodic tuning curves, respectively. Here, we show that the possibility of catastrophic (large) errors creates a trade-off between decoding time and decoding ability. We investigate how decoding time and stimulus dimensionality affect the optimal shape of tuning curves for removing catastrophic errors. In particular, we focus on the spatial periods of the tuning curves for a class of circular tuning curves. We show an overall trend for minimal decoding time to increase with increasing Fisher information, implying a trade-off between accuracy and speed. This trade-off is reinforced whenever the stimulus dimensionality is high, or there is ongoing activity. Thus, given constraints on processing speed, we present normative arguments for the existence of the single-peaked tuning organization observed in early visual areas.

Coding for Large-Scale Distributed Machine Learning.

This article aims to give a comprehensive and rigorous review of the principles and recent development of coding for large-scale distributed machine learning (DML). With increasing data volumes and the pervasive deployment of sensors and computing machines, machine learning has become more distributed. Moreover, the involved computing nodes and data volumes for learning tasks have also increased significantly. For large-scale distributed learning systems, significant challenges have appeared in terms of delay, errors, efficiency, etc. To address the problems, various error-control or performance-boosting schemes have been proposed recently for different aspects, such as the duplication of computing nodes. More recently, error-control coding has been investigated for DML to improve reliability and efficiency. The benefits of coding for DML include high-efficiency, low complexity, etc. Despite the benefits and recent progress, however, there is still a lack of comprehensive survey on this topic, especially for large-scale learning. This paper seeks to introduce the theories and algorithms of coding for DML. For primal-based DML schemes, we first discuss the gradient coding with the optimal code distance. Then, we introduce random coding for gradient-based DML. For primal-dual-based DML, i.e., ADMM (alternating direction method of multipliers), we propose a separate coding method for two steps of distributed optimization. Then coding schemes for different steps are discussed. Finally, a few potential directions for future works are also given.

Generalizations of Talagrand Inequality for Sinkhorn Distance Using Entropy Power Inequality.

The distance that compares the difference between two probability distributions plays a fundamental role in statistics and machine learning. Optimal transport (OT) theory provides a theoretical framework to study such distances. Recent advances in OT theory include a generalization of classical OT with an extra entropic constraint or regularization, called entropic OT. Despite its convenience in computation, entropic OT still lacks sufficient theoretical support. In this paper, we show that the quadratic cost in entropic OT can be upper-bounded using entropy power inequality (EPI)-type bounds. First, we prove an HWI-type inequality by making use of the infinitesimal displacement convexity of the OT map. Second, we derive two Talagrand-type inequalities using the saturation of EPI that corresponds to a numerical term in our expressions. These two new inequalities are shown to generalize two previous results obtained by Bolley et al. and Bai et al. Using the new Talagrand-type inequalities, we also show that the geometry observed by Sinkhorn distance is smoothed in the sense of measure concentration. Finally, we corroborate our results with various simulation studies.

Neural Estimator of Information for Time-Series Data with Dependency.

Novel approaches to estimate information measures using neural networks are well-celebrated in recent years both in the information theory and machine learning communities. These neural-based estimators are shown to converge to the true values when estimating mutual information and conditional mutual information using independent samples. However, if the samples in the dataset are not independent, the consistency of these estimators requires further investigation. This is of particular interest for a more complex measure such as the directed information, which is pivotal in characterizing causality and is meaningful over time-dependent variables. The extension of the convergence proof for such cases is not trivial and demands further assumptions on the data. In this paper, we show that our neural estimator for conditional mutual information is consistent when the dataset is generated with samples of a stationary and ergodic source. In other words, we show that our information estimator using neural networks converges asymptotically to the true value with probability one. Besides universal functional approximation of neural networks, a core lemma to show the convergence is Birkhoff's ergodic theorem. Additionally, we use the technique to estimate directed information and demonstrate the effectiveness of our approach in simulations.

The Convex Information Bottleneck Lagrangian.

The information bottleneck (IB) problem tackles the issue of obtaining relevant compressed representations T of some random variable X for the task of predicting Y. It is defined as a constrained optimization problem that maximizes the information the representation has about the task, I ( T ; Y ) , while ensuring that a certain level of compression r is achieved (i.e., I ( X ; T ) ≤ r ). For practical reasons, the problem is usually solved by maximizing the IB Lagrangian (i.e., L IB ( T ; ß ) = I ( T ; Y ) - ß I ( X ; T ) ) for many values of ß ∈ [ 0 , 1 ] . Then, the curve of maximal I ( T ; Y ) for a given I ( X ; T ) is drawn and a representation with the desired predictability and compression is selected. It is known when Y is a deterministic function of X, the IB curve cannot be explored and another Lagrangian has been proposed to tackle this problem: the squared IB Lagrangian: L sq - IB ( T ; ß sq ) = I ( T ; Y ) - ß sq I ( X ; T ) 2 . In this paper, we (i) present a general family of Lagrangians which allow for the exploration of the IB curve in all scenarios; (ii) provide the exact one-to-one mapping between the Lagrange multiplier and the desired compression rate r for known IB curve shapes; and (iii) show we can approximately obtain a specific compression level with the convex IB Lagrangian for both known and unknown IB curve shapes. This eliminates the burden of solving the optimization problem for many values of the Lagrange multiplier. That is, we prove that we can solve the original constrained problem with a single optimization.

Bounds on the Sum-Rate of MIMO Causal Source Coding Systems with Memory under Spatio-Temporal Distortion Constraints.

In this paper, we derive lower and upper bounds on the OPTA of a two-user multi-input multi-output (MIMO) causal encoding and causal decoding problem. Each user's source model is described by a multidimensional Markov source driven by additive i.i.d. noise process subject to three classes of spatio-temporal distortion constraints. To characterize the lower bounds, we use state augmentation techniques and a data processing theorem, which recovers a variant of rate distortion function as an information measure known in the literature as nonanticipatory ϵ-entropy, sequential or nonanticipative RDF. We derive lower bound characterizations for a system driven by an i.i.d. Gaussian noise process, which we solve using the SDP algorithm for all three classes of distortion constraints. We obtain closed form solutions when the system's noise is possibly non-Gaussian for both users and when only one of the users is described by a source model driven by a Gaussian noise process. To obtain the upper bounds, we use the best linear forward test channel realization that corresponds to the optimal test channel realization when the system is driven by a Gaussian noise process and apply a sequential causal DPCM-based scheme with a feedback loop followed by a scaled ECDQ scheme that leads to upper bounds with certain performance guarantees. Then, we use the linear forward test channel as a benchmark to obtain upper bounds on the OPTA, when the system is driven by an additive i.i.d. non-Gaussian noise process. We support our framework with various simulation studies.

SEK: sparsity exploiting k-mer-based estimation of bacterial community composition.

MOTIVATION: Estimation of bacterial community composition from a high-throughput sequenced sample is an important task in metagenomics applications. As the sample sequence data typically harbors reads of variable lengths and different levels of biological and technical noise, accurate statistical analysis of such data is challenging. Currently popular estimation methods are typically time-consuming in a desktop computing environment. RESULTS: Using sparsity enforcing methods from the general sparse signal processing field (such as compressed sensing), we derive a solution to the community composition estimation problem by a simultaneous assignment of all sample reads to a pre-processed reference database. A general statistical model based on kernel density estimation techniques is introduced for the assignment task, and the model solution is obtained using convex optimization tools. Further, we design a greedy algorithm solution for a fast solution. Our approach offers a reasonably fast community composition estimation method, which is shown to be more robust to input data variation than a recently introduced related method. AVAILABILITY AND IMPLEMENTATION: A platform-independent Matlab implementation of the method is freely available at http://www.ee.kth.se/ctsoftware; source code that does not require access to Matlab is currently being tested and will be made available later through the above Web site.

Secure Source Coding with a Public Helper.

We consider secure multi-terminal source coding problems in the presence of a public helper. Two main scenarios are studied: 1) source coding with a helper where the coded side information from the helper is eavesdropped by an external eavesdropper, 2) triangular source coding with a helper where the helper is considered as a public terminal. We are interested in how the helper can support the source transmission subject to a constraint on the amount of information leaked due to its public nature. We characterize the tradeoff between transmission rate, incurred distortion, and information leakage rate at the helper/eavesdropper in the form of a rate-distortion-leakage region for various classes of problems.