Related papers: Metric Multiview Geometry -- a Catalogue in Low Di…
Dimension-varying linear systems are investigated. First, a dimension-free state space is proposed. A cross dimensional distance is constructed to glue vectors of different dimensions together to form a cross-dimensional topological space.…
We introduce a theory of relative tangency for projective algebraic varieties. The dual variety $X_Z^\vee$ of a variety $X$ relative to a subvariety $Z$ is the set of hyperplanes tangent to $X$ at a point of $Z$. We also introduce the…
We give a systematic approach to constructing non-reduced, locally Cohen-Macaulay schemes with reduced support a smooth projective variety. The hierarchy of such structures includes a lot of information about the underlying variety, its…
In connection with the work of Anscombe, Macpherson, Steinhorn and the present author in [1] we investigate the notion of a multidimensional exact class ($R$-mec), a special kind of multidimensional asymptotic class ($R$-mac) with measuring…
We construct a linear system on a general curve in a totally geodesic subvariety of the moduli space of curves. As a consequence, we obtain rank bounds for totally geodesic subvarieties of dimension at least two. Furthermore, we classify…
We study varieties $\mathcal{A}_n$ arising as equivariant compactifications of the space of $n$ points in $\mathbb{C}$ up to overall translation. We define $\mathcal{A}_n$ and examine its basic geometric properties before constructing an…
We study properties of 3-restricted min-wise independent sets of permutations and prove that the class of these sets is isomorphic to the class of subsets of vertices of multi-dimentional hypercube of certain size with certain pair-wise…
We initiate a study of varieties of minimal degree in weighted projective spaces. We call a weighted projective space $\mathbf{P}(w_0,\dots,w_n)$ divisible if $w_i \mid w_{i+1}$ for all $i$. We provide sharp bounds for when a non-degenerate…
In this paper we introduce new affine algebraic varieties whose points correspond to associative algebras. We show that the algebras within a variety share many important homological properties. In particular, any two algebras in the same…
We introduce a notion of good cohomology for multiple lines in $\mathbb{P}^3$ and we classify multiple lines with good cohomology up to multiplicity 4. In particular, we show that the family of space curves of degree d, not lying on a…
We describe an isomorphism of categories conjectured by Kontsevich. If $M$ and $\widetilde{M}$ are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on $M$ and a suitable version of Fukaya's…
During the last decades, we have witnessed a surge of interests of learning a low-dimensional space with discriminative information from one single view. Even though most of them can achieve satisfactory performance in some certain…
We study the arithmetical ranks and the cohomological dimensions of an infinite class of Cohen-Macaulay varieties of minimal degree. Among these we find, on the one hand, infinitely many set-theoretic complete intersections, on the other…
We theoretically analyze and compare the following five popular multiclass classification methods: One vs. All, All Pairs, Tree-based classifiers, Error Correcting Output Codes (ECOC) with randomly generated code matrices, and Multiclass…
We recall the definition of classical polar varieties, as well as those of affine and projective reciprocal polar varieties. The latter are defined with respect to a non-degenerate quadric, which gives us a notion of orthogonality. In…
We show that the Euclidean distance degree of a real orthogonally invariant matrix variety equals the Euclidean distance degree of its restriction to diagonal matrices. We illustrate how this result can greatly simplify calculations in…
We study the Euclidean Distance degree of algebraic neural network models from the perspective of algebraic geometry. Focusing on shallow networks with two neurons, quadratic activation, and scalar output, we identify the associated…
This paper is devoted to the investigation of selected situations when the computation of projective (and other) equivalences of algebraic varieties can be efficiently solved with the help of finding projective equivalences of finite sets…
During the last decades, learning a low-dimensional space with discriminative information for dimension reduction (DR) has gained a surge of interest. However, it's not accessible for these DR methods to achieve satisfactory performance…
In this paper, the problem of multi-view embedding from different visual cues and modalities is considered. We propose a unified solution for subspace learning methods using the Rayleigh quotient, which is extensible for multiple views,…