Urn models and their applications: an approach to modern discrete probability theory, by Norman L. Johnson and Samuel Kotz. Pp xiii, 402. £16·45. 1977. SBN 0 471 44630 0 (Wiley)

In the most basic application of Ant Colony Optimization (ACO), a set of artificial ants find the shortest path between a source and a destination. Ants deposit pheromone on paths they take, preferring paths that have more pheromone on them. Since shorter paths are traversed faster, more pheromone accumulates on them in a given time, attracting more ants and leading to reinforcement of the pheromone trail on shorter paths. This is a positive feedback process that can also cause trails to persist on longer paths, even when a shorter path becomes available. To counteract this persistence on a longer path, ACO algorithms employ remedial measures, such as using negative feedback in the form of uniform evaporation on all paths. Obtaining high performance in ACO algorithms typically requires fine tuning several parameters that govern pheromone deposition and removal. This paper proposes a new ACO algorithm, called EigenAnt, for finding the shortest path between a source and a destination, based on selective pheromone removal that occurs only on the path that is actually chosen for each trip. We prove that the shortest path is the only stable equilibrium for EigenAnt, which means that it is maintained for arbitrary initial pheromone concentrations on paths, and even when path lengths change with time. The EigenAnt algorithm uses only two parameters and does not require them to be finely tuned. Simulations that illustrate these properties are provided.

Ants find the shortest path: a mathematical proof

Ant Colony Optimization (ACO) is a stochastic local search method that has been inspired by the pheromone trail laying and following behavior of some ant species [1]. Artificial ants in ACO essentially are randomized construction procedures that generate solutions based on (artificial) pheromone trails and heuristic information that are associated to solution components. Since the first ACO algorithm has been proposed in 1991, this algorithmic method has attracted a large number of researchers and in the meantime it has reached a significant level of maturity. In fact, ACO is now a well-established search technique for tackling a wide variety of computationally hard problems.

Ant Colony Optimization

Faced with a choice of paths, an ant chooses a path with a higher concentration of pheromone. Subsequently, it drops pheromone on the path chosen. The reinforcement of the pheromone-following behavior favors the selection of an initially discovered path as the preferred path. This may cause a long path to emerge as the preferred path, were it discovered earlier than a shorter path. However, the shortness of the shorter path offsets some of the pheromone accumulated on the initially discovered longer path. In this paper, we model the trail formation behavior as a generalized Polya urn process. For k equal length paths, we give the distribution of pheromone at any time and highlight its sole dependence on the initial pheromone concentrations on paths. Additionally, we propose a method to incorporate the lengths of paths in the urn process and derive how the pheromone distribution alters on its inclusion. Analytically, we show that it is possible, under certain conditions, to reverse the initial bias that may be present in favor of paths that were discovered prior to the discovery of more efficient (shorter) paths. This addresses the Plasticity-Stability dilemma for ants, by laying out the conditions under which the system will remain stable or become plastic and change the path. Finally, we validate our analysis and results using simulations.

Trail formation in ants. A generalized Polya urn process

Collective foraging in ant colonies is as remarkable in that ants are solving a distributed control and optimization task that is still not fully untravelled. Ants deposit pheromone as they travel, and paths with more pheromone are preferred by succeeding ants. Without any direct communication amongst themselves, ants quickly abandon other trails to concentrate on the shortest one. If the food source moves, or a new path is discovered, ants can still overcome initial bias due to pheromone deposited on an earlier path, and switch to the new path. Nevertheless, important questions remain. How much should be advantage offered by a better path discovered later for the ants to switch to it if an initial preference or bias has already built up on an earlier path ? How much bias is too much ? When multiple food sources are present, ants form multiple trails that seem to optimize overall throughput, if that word may be considered apt. These questions have importance in many real life scenarios. How can information be optimally disseminated ? If a competitors's products or technologies have a large initial bias, then how much gain or advantage should a new product or technology provide in order to cultivate a large following or market share ? What kind of share in the market should you probabilistically expect for the gain you offer ? In the presence of dynamically changing traffic scenarios, how can packets be routed without centralized command and control to maximize network throughput ? This talk summarizes some of our recent work. First, we showed that for traditional ant colony optimization models, beyond a certain threshold, initial bias cannot be overcome. That is, even if a better path is found, ants will not switch from a longer path that has been frequented for a long enough time. Next, motivated by some knowledge of biological ants, we suggest a biologically motivated ant pheromone update model that guarantees convergence to the shortest path regardless of initial bias. The algorithm displays this behaviour even when the food source is moved or path lengths change during foraging. Finally, we examine the utility of such an algorithm in the context of routing in communication networks.

Debugging ants: How ants find the shortest route

An ant deposits pheromone along the path that it travels and is more likely to choose a path with a higher concentration of pheromone. The sensing and dropping of pheromone makes it easy to understand the trail forming behavior of ants. The reinforcement tendency of pheromone following behavior ensures selection of the shortest path from a set of paths. The reinforcement tendency of pheromone following behavior also ensures a biased selection of the initially followed paths over a path, which is shorter but discovered through chance at a later point in time. Under what conditions and limits can this initial bias be reversed? In this paper, we answer this question based on a theoretical analysis of the trail forming behavior of ants. We believe our results to contribute to the overall area of understanding how to build scalable systems that evolve to solve complex problems (e.g. point covering or the traveling salesman problem) without the necessity of central command-and-control.

/pdf/mathematical-modeling-and-convergence-analysis-of-trail-9jhpj22akl.pdf

Mathematical modeling and convergence analysis of trail formation

Compressed Sensing (CS) has emerged as an alternate method to acquire high dimensional signals effectively by exploiting the sparsity assumption. However, owing to non-sparse and non-stationary nature, it is extremely difficult to process Electroencephalograph (EEG) signals using CS paradigm. The success of Bayesian algorithms in recovering non-sparse signals has triggered the research in CS based models for neurophysiological signal processing. In this paper, we address the problem of Temporal Modeling of EEG Signals using Block Sparse Variational Bayes (SVB) Framework. Temporal correlation of EEG signals is modeled blockwise using normal variance scale mixtures parameterized via some random and deterministic parameters. Variational inference is exploited to infer the random parameters and Expectation Maximization (EM) is used to obtain the estimate of deterministic parameters. To validate the framework, we present experimental results for benchmark State Visual Evoked Potential (SSVEP) dataset with 40-target Brain-Computer Interface (BCI) speller using two frequency recognition algorithms viz. Canonical Correlation Analysis (CCA) and L1-regularized Multiway CCA. Results show that the proposed temporal model is highly useful in processing SSVEP-EEG signals irrespective of the recognition algorithms used.

Temporal Modeling of EEG Signals using Block Sparse Variational Bayes Framework

Bayesian Sparse Signal Recovery (SSR) for Multiple Measurement Vectors, when elements of each row of solution matrix are correlated, is addressed in the paper. We propose a standard linear Gaussian observation model and a three-level hierarchical estimation framework, based on Gaussian Scale Mixture (GSM) model with some random and deterministic parameters, to model each row of the unknown solution matrix. This hierarchical model induces heavy-tailed marginal distribution over each row which encompasses several choices of distributions viz. Laplace distribution, Student's t distribution and Jeffery prior. Automatic Relevance Determination (ARD) phenomenon introduces sparsity in the model. It is interesting to see that Block Sparse Bayesian Learning framework is a special case of the proposed framework when induced marginal is Jeffrey prior. Experimental results for synthetic signals are provided to demonstrate its effectiveness. We also explore the possibility of using Multiple Measurement Vectors to model Dynamic Hand Posture Database which consists of sequence of temporally correlated hand posture sequence. It can be seen that by exploiting temporal correlation information present in the successive image samples, the proposed framework can reconstruct the data with less linear random measurements with high fidelity.

Sparse Signal Recovery for Multiple Measurement Vectors with Temporally Correlated Entries: A Bayesian Perspective

The exclusivity of a vast majority of cancer mutations remains poorly understood, despite the availability of large amounts of whole genome and exome sequencing data. In clinical settings, this markedly hinders the identification of the previously uncharacterized deleterious mutations due to the unavailability of matched normal samples. We employed state of the art deep learning algorithms for cross-exome learning of mutational embeddings and demonstrated their utility in sequence based detection of cancer-specific Single Nucleotide Variants (SNVs).

/pdf/deep-learning-discerns-cancer-mutation-exclusivity-eggejt0ss9.pdf

Jayadeva Jayadeva

Papers

Mathematical modeling and convergence analysis of trail formation

Temporal Modeling of EEG Signals using Block Sparse Variational Bayes Framework

Sparse Signal Recovery for Multiple Measurement Vectors with Temporally Correlated Entries: A Bayesian Perspective

Deep learning discerns cancer mutation exclusivity