Related papers: Intuitive norms are Euclidean
A hallmark of human cognition is the ability to continually acquire and distill observations of the world into meaningful, predictive theories. In this paper we present a new mechanism for logical theory acquisition which takes a set of…
The Gaussian Correlation Conjecture states that for any two symmetric, convex sets in n-dimensional space and for any centered, Gaussian measure on that space, the measure of the intersection is greater than or equal to the product of the…
We revisit the task of learning a Euclidean metric from data. We approach this problem from first principles and formulate it as a surprisingly simple optimization problem. Indeed, our formulation even admits a closed form solution. This…
Human cognition spans perception, memory, intuitive judgment, deliberative reasoning, action selection, and social inference, yet these capacities are often explained through distinct computational theories. Here we present a unified…
We study the convex hulls of reachable sets of nonlinear systems with bounded disturbances and uncertain initial conditions. Reachable sets play a critical role in control, but remain notoriously challenging to compute, and existing…
We study the number of facets of the convex hull of n independent standard Gaussian points in d-dimensional Euclidean space. In particular, we are interested in the expected number of facets when the dimension is allowed to grow with the…
We prove completeness of preferential conditional logic with respect to convexity over finite sets of points in the Euclidean plane. A conditional is defined to be true in a finite set of points if all extreme points of the set interpreting…
There are investigated the generalized methods of cognition of the Existing, i.e. everything that is able to influence to the cognizer, and everything differed from the Existing is postulated as indistinguishable from the non-existing and…
Given an ideal $\mathcal{I}$ on $\omega$ and a sequence $x$ in a topological vector space, we let the $\mathcal{I}$-core of $x$ be the least closed convex set containing $\{x_n: n \notin I\}$ for all $I \in \mathcal{I}$. We show two…
We prove a fairly general inequality that estimates the number of lattice points in a ball of positive radius in general position in a Euclidean space. The bound is uniform over lattices induced by a matrix having a bounded operator norm.
In this letter, we provide new insight into Willems et al.'s fundamental lemma by studying the concept of universal inputs. An input is called universal if, when applied to any controllable system, it leads to input-output data that…
Invariants underlying shape inference are elusive: a variety of shapes can give rise to the same image, and a variety of images can be rendered from the same shape. The occluding contour is a rare exception: it has both image salience, in…
In the context of large samples, a small number of individuals might spoil basic statistical indicators like the mean. It is difficult to detect automatically these atypical individuals, and an alternative strategy is using robust…
The gradients of convex functions are expressive models of non-trivial vector fields. For example, Brenier's theorem yields that the optimal transport map between any two measures on Euclidean space under the squared distance is realized as…
We classify rational, irreducible quartic symmetroids in projective 3-space. They are either singular along a line or a smooth conic section, or they have a triple point or a tacnode.
Consider an elliptic curve, defined over the rational numbers, and embedded in projective space. The rational points on the curve are viewed as integer vectors with coprime coordinates. What can be said about a rational point if a bound is…
Geometry constitutes a core set of intuitions present in all humans, regardless of their language or schooling [1]. Could brain's built in machinery for processing geometric information take part in uncertainty representation? For decades…
Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of…
A wedge (i.e., a closed nonempty set in the Euclidean space stable under addition and multiplication with non-negative scalars) induces by a standard way a semi-order (a reflexive and transitive binary relation) in the space. The wedges…
We introduce a remarkable new family of norms on the space of $n \times n$ complex matrices. These norms arise from the combinatorial properties of symmetric functions, and their construction and validation involve probability theory,…