On the Persistence of Clustering Solutions and True Number of Clusters in a Dataset

Typically clustering algorithms provide clustering solutions with prespecified number of clusters. The lack of a priori knowledge on the true number of underlying clusters in the dataset makes it important to have a metric to compare the clustering solutions with different number of clusters. This article quantifies a notion of persistence of clustering solutions that enables comparing solutions with different number of clusters. The persistence relates to the range of data-resolution scales over which a clustering solution persists; it is quantified in terms of the maximum over two-norms of all the associated cluster-covariance matrices. Thus we associate a persistence value for each element in a set of clustering solutions with different number of clusters. We show that the datasets where natural clusters are a priori known, the clustering solutions that identify the natural clusters are most persistent - in this way, this notion can be used to identify solutions with true number of clusters. Detailed experiments on a variety of standard and synthetic datasets demonstrate that the proposed persistence-based indicator outperforms the existing approaches, such as, gap-statistic method, $X$-means, $G$-means, $PG$-means, dip-means algorithms and information-theoretic method, in accurately identifying the clustering solutions with true number of clusters. Interestingly, our method can be explained in terms of the phase-transition phenomenon in the deterministic annealing algorithm, where the number of distinct cluster centers changes (bifurcates) with respect to an annealing parameter

Parameterized MDPs and Reinforcement Learning Problems -- A Maximum Entropy Principle Based Framework

We present a framework to address a class of sequential decision making problems. Our framework features learning the optimal control policy with robustness to noisy data, determining the unknown state and action parameters, and performing sensitivity analysis with respect to problem parameters. We consider two broad categories of sequential decision making problems modelled as infinite horizon Markov Decision Processes (MDPs) with (and without) an absorbing state. The central idea underlying our framework is to quantify exploration in terms of the Shannon Entropy of the trajectories under the MDP and determine the stochastic policy that maximizes it while guaranteeing a low value of the expected cost along a trajectory. This resulting policy enhances the quality of exploration early on in the learning process, and consequently allows faster convergence rates and robust solutions even in the presence of noisy data as demonstrated in our comparisons to popular algorithms such as Q-learning, Double Q-learning and entropy regularized Soft Q-learning. The framework extends to the class of parameterized MDP and RL problems, where states and actions are parameter dependent, and the objective is to determine the optimal parameters along with the corresponding optimal policy. Here, the associated cost function can possibly be non-convex with multiple poor local minima. Simulation results applied to a 5G small cell network problem demonstrate successful determination of communication routes and the small cell locations. We also obtain sensitivity measures to problem parameters and robustness to noisy environment data.Comment: 17 pages, 7 figure

Sparse Linear Regression with Constraints: A Flexible Entropy-based Framework

This work presents a new approach to solve the sparse linear regression problem, i.e., to determine a k-sparse vector w in R^d that minimizes the cost ||y - Aw||^2_2. In contrast to the existing methods, our proposed approach splits this k-sparse vector into two parts -- (a) a column stochastic binary matrix V, and (b) a vector x in R^k. Here, the binary matrix V encodes the location of the k non-zero entries in w. Equivalently, it encodes the subset of k columns in the matrix A that map w to y. We demonstrate that this enables modeling several non-trivial application-specific structural constraints on w as constraints on V. The vector x comprises of the actual non-zero values in w. We use Maximum Entropy Principle (MEP) to solve the resulting optimization problem. In particular, we ascribe a probability distribution to the set of all feasible binary matrices V, and iteratively determine this distribution and the vector x such that the associated Shannon entropy gets minimized, and the regression cost attains a pre-specified value. The resulting algorithm employs homotopy from the convex entropy function to the non-convex cost function to avoid poor local minimum. We demonstrate the efficacy and flexibility of our proposed approach in incorporating a variety of practical constraints, that are otherwise difficult to model using the existing benchmark methods

A Dual System-Level Parameterization for Identification from Closed-Loop Data

This work presents a dual system-level parameterization (D-SLP) method for closed-loop system identification. The recent system-level synthesis framework parameterizes all stabilizing controllers via linear constraints on closed-loop response functions, known as system-level parameters. It was demonstrated that several structural, locality, and communication constraints on the controller can be posed as convex constraints on these system-level parameters. In the current work, the identification problem is treated as a {\em dual} of the system-level synthesis problem. The plant model is identified from the dual system-level parameters associated to the plant. In comparison to existing closed-loop identification approaches (such as the dual-Youla parameterization), the D-SLP framework neither requires the knowledge of a nominal plant that is stabilized by the known controller, nor depends upon the choice of factorization of the nominal plant and the stabilizing controller. Numerical simulations demonstrate the efficacy of the proposed D-SLP method in terms of identification errors, compared to existing closed-loop identification techniques

Closed-Loop Identification of Stabilized Models Using Dual Input-Output Parameterization

This paper introduces a dual input-output parameterization (dual IOP) for the identification of linear time-invariant systems from closed-loop data. It draws inspiration from the recent input-output parameterization developed to synthesize a stabilizing controller. The controller is parameterized in terms of closed-loop transfer functions, from the external disturbances to the input and output of the system, constrained to lie in a given subspace. Analogously, the dual IOP method parameterizes the unknown plant with analogous closed-loop transfer functions, also referred to as dual parameters. In this case, these closed-loop transfer functions are constrained to lie in an affine subspace guaranteeing that the identified plant is \emph{stabilized} by the known controller. Compared with existing closed-loop identification techniques guaranteeing closed-loop stability, such as the dual Youla parameterization, the dual IOP neither requires a doubly-coprime factorization of the controller nor a nominal plant that is stabilized by the controller. The dual IOP does not depend on the order and the state-space realization of the controller either, as in the dual system-level parameterization. Simulation shows that the dual IOP outperforms the existing benchmark methods

Towards Efficient Modularity in Industrial Drying: A Combinatorial Optimization Viewpoint

The industrial drying process consumes approximately 12% of the total energy used in manufacturing, with the potential for a 40% reduction in energy usage through improved process controls and the development of new drying technologies. To achieve cost-efficient and high-performing drying, multiple drying technologies can be combined in a modular fashion with optimal sequencing and control parameters for each. This paper presents a mathematical formulation of this optimization problem and proposes a framework based on the Maximum Entropy Principle (MEP) to simultaneously solve for both optimal values of control parameters and optimal sequence. The proposed algorithm addresses the combinatorial optimization problem with a non-convex cost function riddled with multiple poor local minima. Simulation results on drying distillers dried grain (DDG) products show up to 12% improvement in energy consumption compared to the most efficient single-stage drying process. The proposed algorithm converges to local minima and is designed heuristically to reach the global minimum

Ethics-Relevant Values as Antecedents of Personality Change: Longitudinal Findings from the Life and Time Study

What leads personality to develop in adulthood? Values, guiding principles that apply across contexts, may capture motivation for growth and change. An essentialist trait perspective posits that personality changes only as a result of organic factors. But evidence suggests that psychosocial factors also influence personality change, especially during young adulthood. In the Life and Time study of sources of personality change in adulthood, we specifically explore ethically-relevant value priorities, those related to the relative prioritization of narrow self-interest over the concerns of a larger community. According to Rollo May (1967), “mature values”, including aspects of both self-transcendence and self-determination, should serve to diminish or prevent neurotic anxiety. This is consistent with research on materialism, which is associated with lower well-being. An index based on May’s proposal and several related constructs (materialism, unmitigated self-interest, collectivism and individualism) are tested longitudinally as possible antecedents of Big Five/Six personality trait change using bivariate LCMSR models in a national community sample (N = 864 at Time 1). Contrary to an essentialist trait perspective, these value priorities more often preceded change in personality traits than vice-versa. Somewhat consistent with May’s theory, higher “mature” values preceded higher openness (statistically significant at the p < .005 level). Higher vertical individualism significantly preceded lower compassion, intellect and openness. At the suggestive (p < .05) level, higher unmitigated self-interest preceded lower conscientiousness, higher vertical individualism preceded higher volatility, higher mature values preceded higher honesty/propriety and politeness, higher horizontal collectivism preceded higher orderliness, agreeableness, and assertiveness and lower intellect, and higher horizontal individualism preceded lower withdrawal. In two of three cases, suggestive personality-as-antecedent-of-values-change effects were reciprocal with the values-effects: higher conscientiousness scores reciprocally preceded lower unmitigated self-interest, and higher volatility higher vertical individualism. No significant or suggestive “stand-alone”, non-reciprocal personality on values effects were found