Let a polytope $\mathcal{P}$ be defined by one of the following ways: (i) $\mathcal{P} = \{x \in \mathbb{R}^n \colon A x \leq b\}$, where $A \in \mathbb{Z}^{(n+m) \times n}$, $b \in \mathbb{Z}^{(n+m)}$, and $rank(A) = n$, (ii) $\mathcal{P} = \{x \in \mathbb{R}_+^n \colon A x = b\}$, where $A \in \mathbb{Z}^{m \times n}$, $b \in \mathbb{Z}^{m}$, and $rank(A) = m$, and let all the rank minors of $A$ be bounded by $\Delta$ in the absolute values. We show that $|\mathcal{P} \cap \mathbb{Z}^n|$ can be computed with an algorithm, having the arithmetic complexity bound $$ O\bigl( \nu(d,m,\Delta) \cdot d^3 \cdot \Delta^4 \cdot \log(\Delta) \bigr), $$ where $d = \dim(\mathcal{P})$ and $\nu(d,m,\Delta)$ is the maximal possible number of vertices in a $d$-dimensional polytope $P$, defined by one of the systems above. Using the obtained result, we have the following arithmetical complexity bounds to compute $|P \cap \mathbb{Z}^n|$: 1) The bound $O(\frac{d}{m}+1)^m \cdot d^3 \cdot \Delta^4 \cdot \log(\Delta)$ that is polynomial on $d$ and $\Delta$, for any fixed $m$; 2) The bound $O\bigl(\frac{m}{d}+1\bigr)^{\frac{d}{2}} \cdot d^4 \cdot \Delta^4 \cdot \log(\Delta)$ that is polynomial on $m$ and $\Delta$, for any fixed $d$; 3) The bound $O(d)^{4 + \frac{d}{2}} \cdot \Delta^{4+d} \cdot \log(\Delta)$ that is polynomial on $\Delta$, for any fixed $d$. Given bounds can be used to obtain faster algorithms for the ILP feasibility problem, and for the problem to count integer points in a simplex or in an unbounded Subset-Sum polytope. Unbounded and parametric versions of the above problem are also considered.
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In this paper we prove upper and lower bounds on the minimal spherical dispersion. In particular, we see that the inverse $N(\varepsilon,d)$ of the minimal spherical dispersion is, for fixed $\varepsilon>0$, up to logarithmic terms linear in the dimension $d$. We also derive upper and lower bounds on the expected dispersion for points chosen independently and uniformly at random from the Euclidean unit sphere.
We prove that the border rank of the Kronecker square of the little Coppersmith-Winograd tensor $T_{cw,q}$ is the square of its border rank for $q > 2$ and that the border rank of its Kronecker cube is the cube of its border rank for $q > 4$. This answers questions raised implicitly in [Coppersmith-Winograd, 1990] and explicitly in [Bl\"aser, 2013] and rules out the possibility of proving new upper bounds on the exponent of matrix multiplication using the square or cube of a little Coppersmith-Winograd tensor in this range. In the positive direction, we enlarge the list of explicit tensors potentially useful for Strassen's laser method, introducing a skew-symmetric version of the Coppersmith-Winograd tensor, $T_{skewcw,q}$. For $q = 2$, the Kronecker square of this tensor coincides with the $3\times 3$ determinant polynomial, $\det_3 \in \mathbb{C}^9\otimes \mathbb{C}^9\otimes \mathbb{C}^9$, regarded as a tensor. We show that this tensor could potentially be used to show that the exponent of matrix multiplication is two. We determine new upper bounds for the (Waring) rank and the (Waring) border rank of $\det_3$, exhibiting a strict submultiplicative behaviour for $T_{skewcw,2}$ which is promising for the laser method. We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in $\mathbb{C}^3\otimes \mathbb{C}^3\otimes \mathbb{C}^3$.
Solving the time-dependent Schr\"odinger equation is an important application area for quantum algorithms. We consider Schr\"odinger's equation in the semi-classical regime. Here the solutions exhibit strong multiple-scale behavior due to a small parameter $\hbar$, in the sense that the dynamics of the quantum states and the induced observables can occur on different spatial and temporal scales. Such a Schr\"odinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics. This paper considers quantum analogues of pseudo-spectral (PS) methods on classical computers. Estimates on the gate counts in terms of $\hbar$ and the precision $\varepsilon$ are obtained. It is found that the number of required qubits, $m$, scales only logarithmically with respect to $\hbar$. When the solution has bounded derivatives up to order $\ell$, the symmetric Trotting method has gate complexity $\mathcal{O}\Big({ (\varepsilon \hbar)^{-\frac12} \mathrm{polylog}(\varepsilon^{-\frac{3}{2\ell}} \hbar^{-1-\frac{1}{2\ell}})}\Big),$ provided that the diagonal unitary operators in the pseudo-spectral methods can be implemented with $\mathrm{poly}(m)$ operations. When physical observables are the desired outcomes, however, the step size in the time integration can be chosen independently of $\hbar$. The gate complexity in this case is reduced to $\mathcal{O}\Big({\varepsilon^{-\frac12} \mathrm{polylog}( \varepsilon^{-\frac3{2\ell}} \hbar^{-1} )}\Big),$ with $\ell$ again indicating the smoothness of the solution.
In communication complexity the Arthur-Merlin (AM) model is the most natural one that allows both randomness and non-determinism. Presently we do not have any super-logarithmic lower bound for the AM-complexity of an explicit function. Obtaining such a bound is a fundamental challenge to our understanding of communication phenomena. In this article we explore the gap between the known techniques and the complexity class AM. In the first part we define a new natural class, Small-advantage Layered Arthur-Merlin (SLAM), that has the following properties: - SLAM is (strictly) included in AM and includes all previously known subclasses of AM with non-trivial lower bounds. - SLAM is qualitatively stronger than the union of those classes. - SLAM is a subject to the discrepancy bound: in particular, the inner product function does not have an efficient SLAM-protocol. Structurally this can be summarised as SBP $\cup$ UAM $\subset$ SLAM $\subseteq$ AM $\cap$ PP. In the second part we ask why proving a lower bound of $\omega(\sqrt n)$ on the MA-complexity of an explicit function seems to be difficult. Both of these results are related to the notion of layer complexity, which is, informally, the number of "layers of non-determinism" used by a protocol.
The combinatorial diameter $\operatorname{diam}(P)$ of a polytope $P$ is the maximum shortest path distance between any pair of vertices. In this paper, we provide upper and lower bounds on the combinatorial diameter of a random "spherical" polytope, which is tight to within one factor of dimension when the number of inequalities is large compared to the dimension. More precisely, for an $n$-dimensional polytope $P$ defined by the intersection of $m$ i.i.d.\ half-spaces whose normals are chosen uniformly from the sphere, we show that $\operatorname{diam}(P)$ is $\Omega(n m^{\frac{1}{n-1}})$ and $O(n^2 m^{\frac{1}{n-1}} + n^5 4^n)$ with high probability when $m \geq 2^{\Omega(n)}$. For the upper bound, we first prove that the number of vertices in any fixed two dimensional projection sharply concentrates around its expectation when $m$ is large, where we rely on the $\Theta(n^2 m^{\frac{1}{n-1}})$ bound on the expectation due to Borgwardt [Math. Oper. Res., 1999]. To obtain the diameter upper bound, we stitch these ``shadows paths'' together over a suitable net using worst-case diameter bounds to connect vertices to the nearest shadow. For the lower bound, we first reduce to lower bounding the diameter of the dual polytope $P^\circ$, corresponding to a random convex hull, by showing the relation $\operatorname{diam}(P) \geq (n-1)(\operatorname{diam}(P^\circ)-2)$. We then prove that the shortest path between any ``nearly'' antipodal pair vertices of $P^\circ$ has length $\Omega(m^{\frac{1}{n-1}})$.
Denote by $\Delta_N$ the $N$-dimensional simplex. A map $f\colon \Delta_N\to\mathbb R^d$ is an almost $r$-embedding if $f\sigma_1\cap\ldots\cap f\sigma_r=\emptyset$ whenever $\sigma_1,\ldots,\sigma_r$ are pairwise disjoint faces. A counterexample to the topological Tverberg conjecture asserts that if $r$ is not a prime power and $d\ge2r+1$, then there is an almost $r$-embedding $\Delta_{(d+1)(r-1)}\to\mathbb R^d$. This was improved by Blagojevi\'c-Frick-Ziegler using a simple construction of higher-dimensional counterexamples by taking $k$-fold join power of lower-dimensional ones. We improve this further (for $d$ large compared to $r$): If $r$ is not a prime power and $N:=(d+1)r-r\Big\lceil\dfrac{d+2}{r+1}\Big\rceil-2$, then there is an almost $r$-embedding $\Delta_N\to\mathbb R^d$. For the $r$-fold van Kampen-Flores conjecture we also produce counterexamples which are stronger than previously known. Our proof is based on generalizations of the Mabillard-Wagner theorem on construction of almost $r$-embeddings from equivariant maps, and of the \"Ozaydin theorem on existence of equivariant maps.
Let $CABA$ be the category of complete and atomic boolean algebras and complete boolean homomorphisms, and let $CSL$ be the category of complete meet-semilattices and complete meet-homomorphisms. We show that the forgetful functor from $CABA$ to $CSL$ has a left adjoint. This allows us to describe an endofunctor $H$ on $CABA$ such that the category $Alg(H)$ of algebras for $H$ is dually equivalent to the category $Coalg(\mathcal{P})$ of coalgebras for the powerset endofunctor $\mathcal{P}$ on $Set$. As a consequence, we derive Thomason duality from Tarski duality, thus paralleling how J\'onsson-Tarski duality is derived from Stone duality.
A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In this paper, we consider the problem of approximating functions by polynomials whose Bernstein coefficients with respect to a given degree satisfy such bounds, which implies such bounds on the approximant. We frame the problem as an inequality-constrained optimization problem and give an algorithm for finding the Bernstein coefficients of the exact solution. Additionally, our method can be modified slightly to include equality constraints such as mass preservation. It also extends naturally to multivariate polynomials over a simplex.
Escaping saddle points is a central research topic in nonconvex optimization. In this paper, we propose a simple gradient-based algorithm such that for a smooth function $f\colon\mathbb{R}^n\to\mathbb{R}$, it outputs an $\epsilon$-approximate second-order stationary point in $\tilde{O}(\log n/\epsilon^{1.75})$ iterations. Compared to the previous state-of-the-art algorithms by Jin et al. with $\tilde{O}((\log n)^{4}/\epsilon^{2})$ or $\tilde{O}((\log n)^{6}/\epsilon^{1.75})$ iterations, our algorithm is polynomially better in terms of $\log n$ and matches their complexities in terms of $1/\epsilon$. For the stochastic setting, our algorithm outputs an $\epsilon$-approximate second-order stationary point in $\tilde{O}((\log n)^{2}/\epsilon^{4})$ iterations. Technically, our main contribution is an idea of implementing a robust Hessian power method using only gradients, which can find negative curvature near saddle points and achieve the polynomial speedup in $\log n$ compared to the perturbed gradient descent methods. Finally, we also perform numerical experiments that support our results.
We propose a new method of estimation in topic models, that is not a variation on the existing simplex finding algorithms, and that estimates the number of topics K from the observed data. We derive new finite sample minimax lower bounds for the estimation of A, as well as new upper bounds for our proposed estimator. We describe the scenarios where our estimator is minimax adaptive. Our finite sample analysis is valid for any number of documents (n), individual document length (N_i), dictionary size (p) and number of topics (K), and both p and K are allowed to increase with n, a situation not handled well by previous analyses. We complement our theoretical results with a detailed simulation study. We illustrate that the new algorithm is faster and more accurate than the current ones, although we start out with a computational and theoretical disadvantage of not knowing the correct number of topics K, while we provide the competing methods with the correct value in our simulations.
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