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Polynomial neural networks (PNNs) have been recently shown to be particularly effective at image generation and face recognition, where high-frequency information is critical. Previous studies have revealed that neural networks demonstrate a spectral bias towards low-frequency functions, which yields faster learning of low-frequency components during training. Inspired by such studies, we conduct a spectral analysis of the Neural Tangent Kernel (NTK) of PNNs. We find that the Π-Net family, i.e., a recently proposed parametrization of PNNs, speeds up the learning of the higher frequencies. We verify the theoretical bias through extensive experiments. We expect our analysis to provide novel insights into designing architectures and learning frameworks by incorporating multiplicative interactions via polynomials.

Herein, design of false data injection attack on a distributed cyber-physical system is considered. A stochastic process with linear dynamics and Gaussian noise is measured by multiple agent nodes, each equipped with multiple sensors. The agent nodes form a multi-hop network among themselves. Each agent node computes an estimate of the process by using its sensor observation and messages obtained from neighbouring nodes, via Kalman-consensus filtering. An external attacker, capable of arbitrarily manipulating the sensor observations of some or all agent nodes, injects errors into those sensor observations. The goal of the attacker is to steer the estimates at the agent nodes as close as possible to a pre-specified value, while respecting a constraint on the attack detection probability. To this end, a constrained optimization problem is formulated to find the optimal parameter values of a certain class of linear attacks. The parameters of linear attack are learnt on-line via a combination of stochastic approximation and online stochastic gradient descent. Numerical results demonstrate the efficacy of the attack.The attacker computes S k (t) . = (U k (t)) U k (t) for all 1 ≤ k ≤ N . 5) The sensors make observations {y k (t)} 1≤k≤N , which are accessed by the attacker. 6) The attacker calculates z k (t) = y k (t)−H k Ax (k) (t−1) for all k ∈ {1, 2, · · · , N }. 7) The attacker calculatesz k (t) = T k (t)z k (t) + b k (t) for all k ∈ {1, 2, · · · , N }, where b k (t) ∼ N (M k (t)θ (k) (t− 1) + d k (t), S k (t)) chosen independently of all other

Herein, design of false data injection attack on a distributed cyber-physical system is considered. A stochastic process with linear dynamics and Gaussian noise is measured by multiple agent nodes, each equipped with multiple sensors. The agent nodes form a multi-hop network among themselves. Each agent node computes an estimate of the process by using its sensor observation and messages obtained from neighboring nodes, via Kalman-consensus filtering. An external attacker, capable of arbitrarily manipulating the sensor observations of some or all agent nodes, injects errors into those sensor observations. The goal of the attacker is to steer the estimates at the agent nodes as close as possible to a pre-specified value, while respecting a constraint on the attack detection probability. To this end, a constrained optimization problem is formulated to find the optimal parameter values of a certain class of linear attacks. The parameters of linear attack are learnt on-line via a combination of stochastic approximation based update of a Lagrange multiplier, and an optimization technique involving either the Karush-Kuhn-Tucker (KKT) conditions or online stochastic gradient descent. The problem turns out to be convex for some special cases. Desired convergence of the proposed algorithms are proved by exploiting the convexity and properties of stochastic approximation algorithms. Finally, numerical results demonstrate the efficacy of the attack.

No abstract