Given a statistical model, the maximum likelihood degree is the number of complex solutions to the likelihood equations for generic data. We consider discrete algebraic statistical models and study the solutions to the likelihood equations when the data contain zeros and are no longer generic. Focusing on sampling and model zeros, we show that, in these cases, the solutions to the likelihood equations are contained in a previously studied variety, the likelihood correspondence. The number of these solutions give a lower bound on the ML degree, and the problem of finding critical points to the likelihood function can be partitioned into smaller and computationally easier problems involving sampling and model zeros. We use this technique to compute a lower bound on the ML degree for 2 x 2 x 2 x 2 tensors of border rank ≤ 2 and 3 x n tables of rank ≤ 2 for n = 11, 12, 13, 14, the first four values of n for which the ML degree was previously unknown.
{"title":"Maximum likelihood geometry in the presence of data zeros","authors":"Elizabeth Gross, J. Rodriguez","doi":"10.1145/2608628.2608659","DOIUrl":"https://doi.org/10.1145/2608628.2608659","url":null,"abstract":"Given a statistical model, the maximum likelihood degree is the number of complex solutions to the likelihood equations for generic data. We consider discrete algebraic statistical models and study the solutions to the likelihood equations when the data contain zeros and are no longer generic. Focusing on sampling and model zeros, we show that, in these cases, the solutions to the likelihood equations are contained in a previously studied variety, the likelihood correspondence. The number of these solutions give a lower bound on the ML degree, and the problem of finding critical points to the likelihood function can be partitioned into smaller and computationally easier problems involving sampling and model zeros. We use this technique to compute a lower bound on the ML degree for 2 x 2 x 2 x 2 tensors of border rank ≤ 2 and 3 x n tables of rank ≤ 2 for n = 11, 12, 13, 14, the first four values of n for which the ML degree was previously unknown.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"124 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121403217","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We adapt the rectangular splitting technique of Paterson and Stockmeyer to the problem of evaluating terms in holonomic sequences that depend on a parameter. This approach allows computing the n-th term in a recurrent sequence of suitable type using O(n1/2) "expensive" operations at the cost of an increased number of "cheap" operations. Rectangular splitting has little overhead and can perform better than either naive evaluation or asymptotically faster algorithms for ranges of n encountered in applications. As an example, fast numerical evaluation of the gamma function is investigated. Our work generalizes two previous algorithms of Smith.
{"title":"Evaluating parametric holonomic sequences using rectangular splitting","authors":"Fredrik Johansson","doi":"10.1145/2608628.2608629","DOIUrl":"https://doi.org/10.1145/2608628.2608629","url":null,"abstract":"We adapt the rectangular splitting technique of Paterson and Stockmeyer to the problem of evaluating terms in holonomic sequences that depend on a parameter. This approach allows computing the n-th term in a recurrent sequence of suitable type using O(n1/2) \"expensive\" operations at the cost of an increased number of \"cheap\" operations.\u0000 Rectangular splitting has little overhead and can perform better than either naive evaluation or asymptotically faster algorithms for ranges of n encountered in applications. As an example, fast numerical evaluation of the gamma function is investigated. Our work generalizes two previous algorithms of Smith.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132255972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give an algorithm for solving bivariate polynomial systems over either k(T)[X,Y] or Q[X,Y] using a combination of lifting and modular composition techniques.
{"title":"On the complexity of solving bivariate systems: the case of non-singular solutions","authors":"R. Lebreton, E. Mehrabi, É. Schost","doi":"10.1145/2465506.2465950","DOIUrl":"https://doi.org/10.1145/2465506.2465950","url":null,"abstract":"We give an algorithm for solving bivariate polynomial systems over either <i>k</i>(<i>T</i>)[<i>X,Y</i>] or <i>Q</i>[<i>X,Y</i>] using a combination of lifting and modular composition techniques.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126901316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The F5 algorithm [8] is generally believed as one of the fastest algorithms for computing Gröbner bases. However, its termination problem is still unclear. The crux lies in the non-determinacy of the F5 in selecting which from the critical pairs of the same degree. In this paper, we construct a generalized algorithm F5GEN which contain the F5 as its concrete implementation. Then we prove the correct termination of the F5GEN algorithm. That is to say, for any finite set of homogeneous polynomials, the F5 terminates correctly.
{"title":"The termination of the F5 algorithm revisited","authors":"Senshan Pan, Yu-pu Hu, Baocang Wang","doi":"10.1145/2465506.2465520","DOIUrl":"https://doi.org/10.1145/2465506.2465520","url":null,"abstract":"The F5 algorithm [8] is generally believed as one of the fastest algorithms for computing Gröbner bases. However, its termination problem is still unclear. The crux lies in the non-determinacy of the F5 in selecting which from the critical pairs of the same degree. In this paper, we construct a generalized algorithm F5GEN which contain the F5 as its concrete implementation. Then we prove the correct termination of the F5GEN algorithm. That is to say, for any finite set of homogeneous polynomials, the F5 terminates correctly.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123637486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This tutorial will explain the algorithm behind the currently fastest implementations for univariate factorization over the rationals. The complexity will be analyzed; it turns out that modifications were needed in order to prove a polynomial time complexity while preserving the best practical performance. The complexity analysis leads to two results: (1) it shows that the practical performance on common inputs can be improved without harming the worst case performance, and (2) it leads to an improved complexity, not only for factoring, but for LLL reduction as well.
{"title":"The complexity of factoring univariatepolynomials over the rationals: tutorial abstract","authors":"M. V. Hoeij","doi":"10.1145/2465506.2479779","DOIUrl":"https://doi.org/10.1145/2465506.2479779","url":null,"abstract":"This tutorial will explain the algorithm behind the currently fastest implementations for univariate factorization over the rationals. The complexity will be analyzed; it turns out that modifications were needed in order to prove a polynomial time complexity while preserving the best practical performance.\u0000 The complexity analysis leads to two results: (1) it shows that the practical performance on common inputs can be improved without harming the worst case performance, and (2) it leads to an improved complexity, not only for factoring, but for LLL reduction as well.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130688353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
François Boulier, F. Lemaire, G. Regensburger, M. Rosenkranz
In this paper, we provide a differential algebra algorithm for integrating fractions of differential polynomials. It is not restricted to differential fractions that are the derivatives of other differential fractions. The algorithm leads to new techniques for representing differential fractions, which may help converting differential equations to integral equations (as for example used in parameter estimation).
{"title":"On the integration of differential fractions","authors":"François Boulier, F. Lemaire, G. Regensburger, M. Rosenkranz","doi":"10.1145/2465506.2465934","DOIUrl":"https://doi.org/10.1145/2465506.2465934","url":null,"abstract":"In this paper, we provide a differential algebra algorithm for integrating fractions of differential polynomials. It is not restricted to differential fractions that are the derivatives of other differential fractions. The algorithm leads to new techniques for representing differential fractions, which may help converting differential equations to integral equations (as for example used in parameter estimation).","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"252 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114368391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider a (D x D) symmetric matrix A whose entries are linear forms in Q[X1, ..., Xk] with coefficients of bit size ≤ τ. We provide an algorithm which decides the existence of rational solutions to the linear matrix inequality A ≥ 0 and outputs such a rational solution if it exists. This problem is of first importance: it can be used to compute algebraic certificates of positivity for multivariate polynomials. Our algorithm runs within (k≤)O(1)2O(min(k, D)D2)DO(D2) bit operations; the bit size of the output solution is dominated by τO(1)2O(min(k, D)D2). These results are obtained by designing algorithmic variants of constructions introduced by Klep and Schweighofer. This leads to the best complexity bounds for deciding the existence of sums of squares with rational coefficients of a given polynomial. We have implemented the algorithm; it has been able to tackle Scheiderer's example of a multivariate polynomial that is a sum of squares over the reals but not over the rationals; providing the first computer validation of this counter-example to Sturmfels' conjecture.
{"title":"Computing rational solutions of linear matrix inequalities","authors":"Q. Guo, M. S. E. Din, L. Zhi","doi":"10.1145/2465506.2465949","DOIUrl":"https://doi.org/10.1145/2465506.2465949","url":null,"abstract":"Consider a (D x D) symmetric matrix <b>A</b> whose entries are linear forms in Q[X<sup>1</sup>, ..., X<sub>k</sub>] with coefficients of bit size ≤ τ. We provide an algorithm which decides the existence of rational solutions to the linear matrix inequality <b>A</b> ≥ 0 and outputs such a rational solution if it exists. This problem is of first importance: it can be used to compute algebraic certificates of positivity for multivariate polynomials. Our algorithm runs within (k≤)<sup>O(1)</sup>2<sup>O(min(k, D)</sup>D<sup>2)</sup>D<sup>O</sup>(D<sup>2)</sup> bit operations; the bit size of the output solution is dominated by τ<sup>O(1)</sup>2<sup>O(min(k, D)</sup>D<sup>2)</sup>. These results are obtained by designing algorithmic variants of constructions introduced by Klep and Schweighofer. This leads to the best complexity bounds for deciding the existence of sums of squares with rational coefficients of a given polynomial. We have implemented the algorithm; it has been able to tackle Scheiderer's example of a multivariate polynomial that is a sum of squares over the reals but not over the rationals; providing the first computer validation of this counter-example to Sturmfels' conjecture.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124639559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study integrability conditions for systems of parameterized linear difference equations and related properties of linear differential algebraic groups. We show that isomonodromicity of such a system is equivalent to isomonodromicity with respect to each parameter separately under a linearly differentially closed assumption on the field of differential parameters. Due to our result, it is no longer necessary to solve non-linear differential equations to verify isomonodromicity, which will improve efficiency of computation with these systems. Moreover, it is not possible to further strengthen this result by removing the requirement on the parameters, as we show by giving a counterexample. We also discuss the relation between isomonodromicity and the properties of the associated parameterized difference Galois group.
{"title":"Integrability conditions for parameterized linear difference equations","authors":"Mariya Bessonov, A. Ovchinnikov, M. Shapiro","doi":"10.1145/2465506.2465942","DOIUrl":"https://doi.org/10.1145/2465506.2465942","url":null,"abstract":"We study integrability conditions for systems of parameterized linear difference equations and related properties of linear differential algebraic groups. We show that isomonodromicity of such a system is equivalent to isomonodromicity with respect to each parameter separately under a linearly differentially closed assumption on the field of differential parameters. Due to our result, it is no longer necessary to solve non-linear differential equations to verify isomonodromicity, which will improve efficiency of computation with these systems. Moreover, it is not possible to further strengthen this result by removing the requirement on the parameters, as we show by giving a counterexample. We also discuss the relation between isomonodromicity and the properties of the associated parameterized difference Galois group.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129444414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Critical point methods are at the core of the interplay between polynomial optimization and polynomial system solving over the reals. These methods are used in algorithms for solving various problems such as deciding the existence of real solutions of polynomial systems, performing one-block real quantifier elimination, computing the real dimension of the solution set, etc. The input consists of $s$ polynomials in $n$ variables of degree at most $D$. Usually, the complexity of the algorithms is $(s, D)^{O(n^alpha)}$ where $alpha$ is a constant. In the past decade, tremendous efforts have been deployed to improve the exponents in the complexity bounds. This led to efficient implementations and new geometric procedures for solving polynomial systems over the reals that exploit properties of critical points. In this talk, we present an overview of these techniques and their impact on practical algorithms. Also, we show how we can tune them to exploit algebraic and geometric structures in two fundamental problems. The first one is real root finding of determinants of $n$-variate linear matrices of size $ktimes k$. We introduce an algorithm whose complexity is polynomial in ${{n+k}choose{k}}$ (joint work with S. Naldi and D. Henrion). This improves the previously known $k^{O(n)}$ bound. The second one is about computing the real dimension of a semi-algebraic set. We present a probabilistic algorithm with complexity $(s, D)^{O(n)}$, that improves the long-standing $(s, D)^{O(n^2)}$ bound obtained by Koi-ran (joint work with E. Tsigaridas).
{"title":"Critical point methods and effective real algebraic geometry: new results and trends","authors":"M. S. E. Din","doi":"10.1145/2465506.2465928","DOIUrl":"https://doi.org/10.1145/2465506.2465928","url":null,"abstract":"Critical point methods are at the core of the interplay between polynomial optimization and polynomial system solving over the reals. These methods are used in algorithms for solving various problems such as deciding the existence of real solutions of polynomial systems, performing one-block real quantifier elimination, computing the real dimension of the solution set, etc. The input consists of $s$ polynomials in $n$ variables of degree at most $D$. Usually, the complexity of the algorithms is $(s, D)^{O(n^alpha)}$ where $alpha$ is a constant. In the past decade, tremendous efforts have been deployed to improve the exponents in the complexity bounds. This led to efficient implementations and new geometric procedures for solving polynomial systems over the reals that exploit properties of critical points. In this talk, we present an overview of these techniques and their impact on practical algorithms. Also, we show how we can tune them to exploit algebraic and geometric structures in two fundamental problems. The first one is real root finding of determinants of $n$-variate linear matrices of size $ktimes k$. We introduce an algorithm whose complexity is polynomial in ${{n+k}choose{k}}$ (joint work with S. Naldi and D. Henrion). This improves the previously known $k^{O(n)}$ bound. The second one is about computing the real dimension of a semi-algebraic set. We present a probabilistic algorithm with complexity $(s, D)^{O(n)}$, that improves the long-standing $(s, D)^{O(n^2)}$ bound obtained by Koi-ran (joint work with E. Tsigaridas).","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"34 4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128601916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
I last spoke at a computer algebra conference in August 1981. Since that time I created Mathematica (launched 1988) andWolfram|Alpha (launched 2009). This talk will survey perspectives on computer algebra gained through these activities, as well as through my work in basic science. I will also describe what I see as being key future directions and aspirations for computer algebra.
{"title":"Computer algebra: a 32-year update","authors":"S. Wolfram","doi":"10.1145/2465506.2465930","DOIUrl":"https://doi.org/10.1145/2465506.2465930","url":null,"abstract":"I last spoke at a computer algebra conference in August 1981. Since that time I created Mathematica (launched 1988) andWolfram|Alpha (launched 2009). This talk will survey perspectives on computer algebra gained through these activities, as well as through my work in basic science. I will also describe what I see as being key future directions and aspirations for computer algebra.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130295597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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