arXiv subjects
Publications and source records attributed to Mateusz Michalek.
We provide a complete description of the ideal that serves as the resultant ideal for n univariate polynomials of degree d. We in particular describe a set of generators of this resultant ideal arising as maximal minors of a set of cascading matrices formed from the coefficients of the polynomials, generalising the classical Sylvester resultant of two polynomials.
By the work of J.Huh, one can interpret binomial coefficients as a solution to an intersection problem on a permutohedral variety $X_E$. Applying Hirzebruch-Riemann-Roch, this intersection problem is equivalent to computing Euler characteristic of a specific element of $K$-theory of $X_E$. This element has a natural lifting to equivariant $K$-theory and thus the Euler characteristic may be upgraded to a Laurent polynomial. We provide and implement three different approaches, in particular a recursive one, to computing these polynomials.
We show that the empirical risk minimization (ERM) problem for neural networks has no solution in general. Given a training set $s_1, \dots, s_n \in \mathbb{R}^p$ with corresponding responses $t_1,\dots,t_n \in \mathbb{R}^q$, fitting a $k$-layer neural network $ν_θ: \mathbb{R}^p \to \mathbb{R}^q$ involves estimation of the weights $θ\in \mathbb{R}^m$ via an ERM: \[ \inf_{θ\in \mathbb{R}^m} \; \sum_{i=1}^n \lVert t_i - ν_θ(s_i) \rVert_2^2. \] We show that even for $k = 2$, this infimum is not attainable in general for common activations like ReLU, hyperbolic tangent, and sigmoid functions. A high-level explanation is like that for the nonexistence of best rank-$r$ approximations of higher-order tensors --- the set of parameters is not a closed set --- but the geometry involved for best $k$-layer neural networks approximations is more subtle. In addition, we show that for smooth activations $σ(x)= 1/\bigl(1 + \exp(-x)\bigr)$ and $σ(x)=\tanh(x)$, such failure to attain an infimum can happen on a positive-measured subset of responses. For the ReLU activation $σ(x)=\max(0,x)$, we completely classifying cases where the ERM for a best two-layer neural network approximation attains its infimum. As an aside, we obtain a precise description of the geometry of the space of two-layer neural networks with $d$ neurons in the hidden layer: it is the join locus of a line and the $d$-secant locus of a cone.
Motivated by questions of Mulmuley and Stanley we investigate quasi-polynomials arising in formulas for plethysm. We demonstrate, on the examples of $S^3(S^k)$ and $S^k(S^3)$, that these need not be counting functions of inhomogeneous polytopes of dimension equal to the degree of the quasi-polynomial. It follows that these functions are not, in general, counting functions of lattice points in any scaled convex bodies, even when restricted to single rays. Our results also apply to special rectangular Kronecker coefficients.
We apply lattice point counting methods to compute the multiplicities in the plethysm of $GL(n)$. Our approach gives insight into the asymptotic growth of the plethysm and makes the problem amenable to computer algebra. We prove an old conjecture of Howe on the leading term of plethysm. For any partition $μ$ of 3,4, or 5 we obtain an explicit formula in $λ$ and $k$ for the multiplicity of $S^λ$ in $S^μ(S^k)$.
Group-based models arise in algebraic statistics while studying evolution processes. They are represented by embedded toric algebraic varieties. Both from the theoretical and applied point of view one is interested in determining the ideals defining the varieties. Conjectural bounds on the degree in which these ideals are generated were given by Sturmfels and Sullivant. We prove that for the 3-Kimura model, corresponding to the group G=Z2xZ2, the projective scheme can be defined by an ideal generated in degree 4. In particular, it is enough to consider degree 4 phylogenetic invariants to test if a given point belongs to the variety. We also investigate G-models, a generalization of abelian group-based models. For any G-model, we prove that there exists a constant $d$, such that for any tree, the associated projective scheme can be defined by an ideal generated in degree at most d.
We study the secant line variety of the Segre product of projective spaces using special cumulant coordinates adapted for secant varieties. We show that the secant variety is covered by open normal toric varieties. We prove that in cumulant coordinates its ideal is generated by binomial quadrics. We present new results on the local structure of the secant variety. In particular, we show that it has rational singularities and we give a description of the singular locus. We also classify all secant varieties that are Gorenstein. Moreover, generalizing (Sturmfels and Zwiernik 2012), we obtain analogous results for the tangential variety.
In this paper we investigate properties of algebraic varieties representing group-based phylogenetic models. We propose a method of generating many phylogenetic invariants. We prove that we obtain all invariants for any tree for the binary Jukes-Cantor model. We conjecture that our method can give all phylogenetic invariants for any tree. We show that for 3-Kimura our conjecture is equivalent to the conjecture of Sturmfels and Sullivant. This, combined with the results of Sturmfels and Sullivant, would make it possible to determine all phylogenetic invariants for any tree for 3-Kimura model, and also other phylogenetic models. Next we give the (first) example of a non-normal general group-based model for an abelian group. Following Kubjas we also determine some invariants of group-based models showing that the associated varieties do not have to be deformation equivalent.
We give a short proof of Weintraub's conjecture by constructing explicit highest weight vectors in the symmetric power of an even exterior power.
In this paper we present geometric features of group based models. We focus on the 3-Kimura model. We present a precise geometric description of the variety associated to any tree on a Zariski open set. In particular this set contains all biologically meaningful points. Our motivation is a conjecture of Sturmfels and Sullivant on the degree in which the ideal associated to 3-Kimura model is generated.
In this short note we present a family of counterexamples to the King's conjecture.
In this paper we complete the results of Sullivant and Sturmfels proving that many of the algebraic group-based models for Markov processes on trees are pseudo-toric. We also show in which cases these varieties are normal. This is done by the generalization of the discrete Fourier transform approach. In the next step, following Sullivant and Sturmfels, we describe a fast algorithm finding a polytope associated to these algebraic models. However in our case we apply the notions of sockets and networks extending the work of Buczynska and Wisniewski who introduced it for the binary case.
We investigate full strongly exceptional collections on smooth, com- plete toric varieties. We obtain explicit results for a large family of varieties with Picard number three, containing many of the families already known. We also describe the relations between the collections and the split of the push forward of the trivial line bundle by the toric Frobenius morphism.
We construct a full, strongly exceptional collection of line bundles on the variety X that is the blow up of the projectivization of the vector bundle O_{P^{n-1}}\oplus O_{P^{n-1}}(b) along a linear space of dimension n-2, where b is a non-negative integer.
We consider two varieties associated to a web of quadrics W in the projective space of dimension 7. One is the base locus and the second one is the double cover of the three dimensional projective space branched along the determinant surface of W. We show that small resolutions of these varieties are Calabi-Yau manifolds. We compute their Betti numbers and show that they are not birational in the generic case. The main result states that if the base locus of W contains a plane then in the generic case the two varieties are birational.
In this note we give a short, direct proof of the well known Combinatorial Nullstellensatz.