Collision-free motion is a fundamental requirement for many autonomous systems. This paper develops a safety-critical control approach for the collision-free navigation of polytope-shaped agents in polytope-shaped environments. A systematic method is proposed to generate control barrier function candidates in closed form that lead to controllers with formal safety guarantees. The proposed approach is demonstrated through simulation, with obstacle avoidance examples in 2D and 3D, including dynamically changing environments.

Concentration of measure has been argued to be the fundamental cause of adversarial vulnerability. Mahloujifar et al. presented an empirical way to measure the concentration of a data distribution using samples, and employed it to find lower bounds on intrinsic robustness for several benchmark datasets. However, it remains unclear whether these lower bounds are tight enough to provide a useful approximation for the intrinsic robustness of a dataset. To gain a deeper understanding of the concentration of measure phenomenon, we first extend the Gaussian Isoperimetric Inequality to non-spherical Gaussian measures and arbitrary $\ell_p$-norms ($p \geq 2$). We leverage these theoretical insights to design a method that uses half-spaces to estimate the concentration of any empirical dataset under $\ell_p$-norm distance metrics. Our proposed algorithm is more efficient than Mahloujifar et al.'s, and our experiments on synthetic datasets and image benchmarks demonstrate that it is able to find much tighter intrinsic robustness bounds. These tighter estimates provide further evidence that rules out intrinsic dataset concentration as a possible explanation for the adversarial vulnerability of state-of-the-art classifiers.

In it we said the perceptron takes some input value \([x_1, x_2, ..., x_i, ..., x_n]\), also called features, some weights \([w_1, w_2, ..., w_i, ..., w_n]\), multiplies them with each other and takes the sum of these multiplications: This is the definition of a Linear Combination: it is the sum of some terms multiplied by constant values. In our case the terms are the features and the constants are the weights.

Recently dictionary screening has been proposed as an effective way to improve the computational efficiency of solving the lasso problem, which is one of the most commonly used method for learning sparse representations. To address today's ever increasing large dataset, effective screening relies on a tight region bound on the solution to the dual lasso. Typical region bounds are in the form of an intersection of a sphere and multiple half spaces. One way to tighten the region bound is using more half spaces, which however, adds to the overhead of solving the high dimensional optimization problem in lasso screening. This paper reveals the interesting property that the optimization problem only depends on the projection of features onto the subspace spanned by the normals of the half spaces. This property converts an optimization problem in high dimension to much lower dimension, and thus sheds light on reducing the computation overhead of lasso screening based on tighter region bounds.

Interest in this algorithm waned in the 1970's after it was emphasized[Minsky and Papert, 1969] (1) that the class of problems solvable by a single half space was limited, and (2) that the Perceptron algorithm, although converging in finite time, did not converge in polynomial time. In the 1980's, however, it has become evident that there is no hope of providing a learning algorithm which can learn arbitrary functions in polynomial time and much research has thus been restricted to algorithms which learn a function drawn from a particular class of functions. Moreover, learning theory has focused on protocols like that of [Valiant, 1984] where we seek to classify, not a fixed set of examples, but examples drawn from a probability distribution. This allows a natural notion of "generalization". There are very few classes which have yet been proven learnable in polynomial time, and one of these is the class of half spaces. Thus there is considerable theoretical interest now in studying the problem of learning a single half space, and so it is natural to reexamine the Percept ron algorithm within the formalism of Valiant.

Interest in this algorithm waned in the 1970's after it was emphasized[Minsky and Papert, 1969] (1) that the class of problems solvable by a single half space was limited, and (2) that the Perceptron algorithm, although converging in finite time, did not converge in polynomial time. In the 1980's, however, it has become evident that there is no hope of providing a learning algorithm which can learn arbitrary functions in polynomial time and much research has thus been restricted to algorithms which learn a function drawn from a particular class of functions. Moreover, learning theory has focused on protocols like that of [Valiant, 1984] where we seek to classify, not a fixed set of examples, but examples drawn from a probability distribution. This allows a natural notion of "generalization". There are very few classes which have yet been proven learnable in polynomial time, and one of these is the class of half spaces. Thus there is considerable theoretical interest now in studying the problem of learning a single half space, and so it is natural to reexamine the Percept ron algorithm within the formalism of Valiant.

Interest in this algorithm waned in the 1970's after it was emphasized[Minsky andPapert, 1969] (1) that the class of problems solvable by a single half space was limited, and (2) that the Perceptron algorithm, although converging infinite time, did not converge in polynomial time. In the 1980's, however, it has become evident that there is no hope of providing a learning algorithm which can learn arbitrary functions in polynomial time and much research has thus been restricted to algorithms which learn a function drawn from a particular class of functions. Moreover, learning theory has focused on protocols like that of [Valiant, 1984] where we seek to classify, not a fixed set of examples, but examples drawn from a probability distribution. This allows a natural notion of "generalization" . There are very few classes which have yet been proven learnable in polynomial time, and one of these is the class of half spaces. Thus there is considerable theoretical interest now in studying the problem of learning a single half space, and so it is natural to reexamine the Percept ron algorithm within the formalism of Valiant. The Perceptron Algorithm Is Fast for Non-Malicious Distributions 677 In Valiant's protocol, a class of functions is called learnable if there is a learning algorithm which works in polynomial time independent of the distribution D generating the examples. Under this definition the Perceptron learning algorithm is not a polynomial time learning algorithm. However we will argue in section 2 that this definition is too restrictive.