In this work, we introduce a family of novel activation functions for deep neural networks that approximate n-ary, or n-argument, probabilistic logic. Logic has long been used to encode complex relationships between claims that are either true or false. Thus, these activation functions provide a step towards models that can efficiently encode information. Unfortunately, typical feedforward networks with elementwise activation functions cannot capture certain relationships succinctly, such as the exclusive disjunction (p xor q) and conditioned disjunction (if c then p else q). Our n-ary activation functions address this challenge by approximating belief functions (probabilistic Boolean logic) with logit representations of probability and experiments demonstrate the ability to learn arbitrary logical ground truths in a single layer. Further, by representing belief tables using a basis that associates the number of nonzero parameters with the effective arity of each belief function, we forge a concrete relationship between logical complexity and sparsity, thus opening new optimization approaches to suppress logical complexity during training. We provide a computationally efficient PyTorch implementation and test our activation functions against other logic-approximating activation functions on both traditional machine learning tasks as well as reproducing known logical relationships.

Entrepreneurship ecosystem and entrepreneurial behaviour & attitudes are significantly associated with several variables associated with social development, economic development, science & technological development, intellectual property right regime, geographical location, government policies, demographical profile, and cultural aspects of the people. This study creates entrepreneurship ecosystem development index (EEDI) and entrepreneurial behaviour & attitude index (EBAI) to explain the relative performance of entrepreneurship ecosystem and entrepreneurial behaviour & attitudes, respectively. Different indicators associated with entrepreneurship ecosystem and entrepreneurial behaviour & attitudes, respectively used to create EEDI and EBAI. These indicators are recognized as a key element of entrepreneurship ecosystem and entrepreneurial behaviour & attitudes by global entrepreneurship monitor (GEM). Subsequently, it examined the association of estimated EEDI and EBAI by including some relevant indicators as control variables in proposed empirical models.

In this paper, we study shared cache coded caching (SC-CC): a set of caches serves a larger set of users; each user access one cache, and a cache may serve many users. For this problem, under uncoded placement, Parrinello, Ünsal, and Elia showed an optimal SC-CC scheme, in which the subpacketization level depends upon the number of caches. We show an SC-CC scheme where the subpacketization level does not directly depend upon the number of users or caches; any number of caches and users can be accommodated for a fixed subpacketization level. We show that given an upper limit on the allowable subpacketization level, our SC-CC scheme may achieve a lesser rate than other relevant SC-CC schemes. Our scheme is constructed using matrices and designs.

We study the problem of multi-access coded caching (MACC): a central server has N files, K (K ≤ N) caches each of which stores M out of the N files, K users each of which demands one out of the N files, and each user accesses z caches. The objective is to jointly design the placement, delivery, and user-to-cache association, to optimize the achievable rate. This problem has been extensively studied in the literature under the assumption that a user accesses only one cache. However, when a user accesses one or more caches, this problem has been studied only under the assumption that a user accesses z consecutive caches with a cyclic wrap-around over the boundaries. A natural question is how other user-to-cache associations fare against the cyclic wrap-around user-to-cache association. A bipartite graph can describe a general user-to-cache association. We identify a class of bipartite graphs that, when used as a user-to-cache association, achieves either a lesser rate or a lesser subpacketization than all other existing MACC schemes using a cyclic wrap-around user-to-cache association. The placement and delivery strategy of our MACC scheme is constructed using a combinatorial structure called maximal cross resolvable design.

Tracking objects in Computer Vision is a hard problem. Privacy and utility concerns adds an extra layer of complexity over this problem. In this work we consider the problem of maintaining privacy and utility while tracking an object in a video stream using Kalman filtering. Our first proposed method ensures that the localization accuracy of this object will not improve beyond a certain level. Our second method ensures that the localization accuracy of the same object will always remain under a certain threshold.

Vector linear solvability of non-multicast networks depends upon both the characteristic of the finite held and the dimension of the vector linear network code. In the literature, the dependency on the characteristic of the finite held and the dependency on the dimension have been studied separately. In this paper, we show the interdependency between the characteristic of the finite held and the dimension of the vector linear network code that achieves a vector linear network coding (VLNC) solution in non-multicast networks. For any given network Al, we dehne P(N, d) as the set of all characteristics of finite fields over which the network N has a d-dimensional VLNC solution. To the best of our knowledge, for any network N shown in the literature, if P(N, 1) is non-empty, then P(N, 1) = P(N, d) for any positive integer d. We show that, for any two non-empty sets of primes P1and P2, there exists a network N such that P(N, 1) = P1, but P(N, 2) = {P1, P2}. We also show that there are networks exhibiting a similar advantage (the existence of a VLNC solution over a larger set of characteristics) if the dimension is increased from 2 to 3. However, such behaviour is not universal, as there exist networks which admit a VLNC solution over a smaller set of characteristics of finite fields when the dimension is increased. Using the networks constructed in this paper, we further demonstrate that: (i) a network having an m1-dimensional VLNC solution over a finite held of some characteristic and an m2-dimensional VLNC solution over a finite held of some other characteristic may not have an (m1+ m2)-dimensional VLNC solution over any finite held; (ii) there exist a class of networks for which scalar linear network coding (SLNC) over non-commutative rings has some advantage over SLNC over finite fields: the least sized non-commutative ring over which each network in the class has an SLNC solution is significantly lesser in size than the least sized finite held over which it has an SLNC solution.

This paper presents yet another instance of the power of vector linear network coding over scalar linear network coding. Previous works have established that the size of the finite field required to achieve a vector linear solution may be smaller than that size of the finite field required to achieve a scalar linear solution. It has been also shown there exist networks which do not have a scalar linear solution but have a vector linear solution. In this paper we show that the set of characteristics over which a network has a vector linear solution may be larger than the set of characteristics over which it has a scalar linear solution. We prove this result by showing a network which has a scalar linear solution if and only if the characteristic of the finite field is 2, but has a 2-dimensional vector linear solution over every finite fields.

Kinematic motor behaviour of robots can be encoded using dynamical systems. These dynamical systems are learnt from human demonstrations to generalize human like motions. The size of the demonstration space involving a robot usually being large, it is not possible to provide all the demonstrations for robot's learning. Hence, it requires an efficient learning architecture that is able to generalize for unseen contexts. In the proposed algorithm, we model the movements, shown in the human demonstrations, as non-linear multivariate dynamics using mixture of Gaussians. Generally, in non-linear multivariate modelling approach pertaining to programming by demonstration, requires structured demonstrations i.e. it always requires to have a fixed and unique equilibrium point during the learning phase. The proposed method can relax these constraints and has the following advantage over the existing work: first, it would be possible to learn from any demonstration which is not constrained to have always the same equilibrium point; second, it would be possible to capture the variations in movement patterns depending upon the position of the equilibrium point. The proposed algorithm has been implemented using Barrett WAM and experimental results have been compared with existing approach.

We consider directed acyclic networks where each terminal requires sum of all the sources. Such a class of networks has been termed as sum-networks in the literature. A sum-network having m sources and n terminals has been termed as a ms/nt sum-network. There has been previous works on the capacity of sum-networks, specifically, it has been shown that the capacity of a 3s/3t sum-network is either 0,2/3 or ≥ 1. In this paper, we consider some generalizations of 3s/3t sum-networks, namely, ms/3t and 3s/nt sum-networks, where m, n ≥ 3. For ms/3t and 3s/nt sum-networks, where m, n ≥ 3, if the mincut between each source and each terminal is at least 1, the capacity is known to be at least 2/3. In this paper, we show that there exist ms/3t and 3s/nt sum-networks whose capacities lie between 2/3 and 1. Specifically, we show that for any positive integer k ≥ 2, there exists a ms/3t sum-network (and also a 3s/nt sum-network) whose capacity is k/k+1. We conjecture that the capacity of a ms/3t sum-network, where m > 3 (and also of a 3s/nt sum-network, where n > 3) is either 0, ≥ 1 or of the form k/k+1, where k is a positive integer greater than or equal to 2.