arXiv subjects
Publications and source records attributed to Sumit Kumar Jha.
In this brief note, we give two explicit formulas for the Bernoulli Numbers in terms of the Stirling numbers of the second kind, and the Eulerian Numbers. To the best of our knowledge, these formulas are new. We also derive two more probably known formulas.
Attribution methods have been developed to explain the decision of a machine learning model on a given input. We use the Integrated Gradient method for finding attributions to define the causal neighborhood of an input by incrementally masking high attribution features. We study the robustness of machine learning models on benign and adversarial inputs in this neighborhood. Our study indicates that benign inputs are robust to the masking of high attribution features but adversarial inputs generated by the state-of-the-art adversarial attack methods such as DeepFool, FGSM, CW and PGD, are not robust to such masking. Further, our study demonstrates that this concentration of high-attribution features responsible for the incorrect decision is more pronounced in physically realizable adversarial examples. This difference in attribution of benign and adversarial inputs can be used to detect adversarial examples. Such a defense approach is independent of training data and attack method, and we demonstrate its effectiveness on digital and physically realizable perturbations.
We calculate asymptotic expansions for the moments of number of comparisons used by the randomized quick sort algorithm using the singularity analysis of certain generating functions.
We establish asymptotic expansions for factorial moments of following distributions: number of cycles in a random permutation, number of inversions in a random permutation, and number of comparisons used by the randomized quick sort algorithm.To achieve this we use singularity analysis of certain type of generating functions due to Flajolet and Odlyzko.
The in-situ permutation algorithm due to MacLeod replaces $(x_{1},\cdots,x_{n})$ by $(x_{p(1)},\cdots,x_{p(n)})$ where $π=(p(1),\cdots,p(n))$ is a permutation of $\{1,2,\cdots,n\}$ using at most $O(1)$ space. Kirshenhofer, Prodinger and Tichy have shown that the major cost incurred in the algorithm satisfies a recurrence similar to sequence of the number of key comparisons needed by the Quicksort algorithm to sort an array of $n$ randomly permuted items. Further, Hwang has proved that the normalized cost converges in distribution. Here, following Neininger and Rüschendorf, we prove the that rate of convergence to be of the order $Θ(\ln(n)/n)$ in the Zolotarev metric.
Let $f:\{-1,1\}^{n}\rightarrow \{-1,1\}$ be a Boolean valued function having total degree $d$. Then a conjecture due to Servedio and Gopalan asserts that $\sum_{i=1}^{n}\widehat{f}(i)\leq \sum_{j=1}^{d}\widehat{\text{Maj}}_{d}(j)$ where $\text{Maj}_{d}$ is the majority function on $d$ bits. Here we give some alternative formalisms of this conjecture involving the discrete derivative operators on $f$.
We revisit the method of Kirschenhofer, Prodinger and Tichy to calculate the moments of comparisons used by the quick sort algorithm. We reemphasize that this approach helps in calculating these quantities with less computation. We also point out that as observed by Knuth this method also gives moments for total path length of a binary search tree built over a random set of n keys.
The formal verification of large probabilistic models is important and challenging. Exploiting the concurrency that is often present is one way to address this problem. Here we study a restricted class of asynchronous distributed probabilistic systems in which the synchronizations determine the probability distribution for the next moves of the participating agents. The key restriction we impose is that the synchronizations are deterministic, in the sense that any two simultaneously enabled synchronizations must involve disjoint sets of agents. As a result, this network of agents can be viewed as a succinct and distributed presentation of a large global Markov chain. A rich class of Markov chains can be represented this way. We define an interleaved semantics for our model in terms of the local synchronization actions. The network structure induces an independence relation on these actions, which, in turn, induces an equivalence relation over the interleaved runs in the usual way. We construct a natural probability measure over these equivalence classes of runs by exploiting Mazurkiewicz trace theory and the probability measure space of the associated global Markov chain. It turns out that verification of our model, called DMCs (distributed Markov chains), can often be efficiently carried out by exploiting the partial order nature of the interleaved semantics. To demonstrate this, we develop a statistical model checking (SMC) procedure and use it to verify two large distributed probabilistic networks.
This paper presents the design of a novel distributed algorithm d-IRA for the reachability analysis of linear hybrid automata. Recent work on iterative relaxation abstraction (IRA) is leveraged to distribute the computational problem among multiple computational nodes in a non-redundant manner by performing careful infeasibility analysis of linear programs corresponding to spurious counterexamples. The d-IRA algorithm is resistant to failure of multiple computational nodes. The experimental results provide promising evidence for the possible successful application of this technique.
This draft suggests a new counterexample guided abstraction refinement (CEGAR) framework that uses the combination of numerical simulation for nonlinear differential equations with linear programming for linear hybrid automata (LHA) to perform reachability analysis on nonlinear hybrid automata. A notion of $ε-$ structural robustness is also introduced which allows the algorithm to validate counterexamples using numerical simulations. Keywords: verification, model checking, hybrid systems, hybrid automata, robustness, robust hybrid systems, numerical simulation, cegar, abstraction refinement.