Nicolás Botbol, Laurent Bus'e, M. Chardin, Fatmanur Yildirim
We contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three dimensional projective space. This problem is first turned into the computation of the finite fibers of a generically finite dominant rational map: a congruence of normal lines to the rational surface. Then, an in-depth study of certain syzygy modules associated to such a congruence is presented and applied to build elimination matrices that provide universal representations of its finite fibers, under some genericity assumptions. These matrices depend linearly in the variables of the three dimensional space. They can be pre-computed so that the orthogonal projections of points are approximately computed by means of fast and robust numerical linear algebra calculations.
{"title":"Fibers of Multi-Graded Rational Maps and Orthogonal Projection onto Rational Surfaces","authors":"Nicolás Botbol, Laurent Bus'e, M. Chardin, Fatmanur Yildirim","doi":"10.1137/19m1289522","DOIUrl":"https://doi.org/10.1137/19m1289522","url":null,"abstract":"We contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three dimensional projective space. This problem is first turned into the computation of the finite fibers of a generically finite dominant rational map: a congruence of normal lines to the rational surface. Then, an in-depth study of certain syzygy modules associated to such a congruence is presented and applied to build elimination matrices that provide universal representations of its finite fibers, under some genericity assumptions. These matrices depend linearly in the variables of the three dimensional space. They can be pre-computed so that the orthogonal projections of points are approximately computed by means of fast and robust numerical linear algebra calculations.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"24 1","pages":"322-353"},"PeriodicalIF":1.2,"publicationDate":"2019-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84593933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide a reduction of the Ring-LWE problem to Ring-LWE problems in subrings, in the presence of samples of a restricted form (i.e. $(a,b)$ such that $a$ is restricted to a multiplicative coset of the subring). To create and exploit such restricted samples, we propose Ring-BKW, a version of the Blum-Kalai-Wasserman algorithm which respects the ring structure. Off-the-shelf BKW dimension reduction (including coded-BKW and sieving) can be used for the reduction phase. Its primary advantage is that there is no need for back-substitution, and the solving/hypothesis-testing phase can be parallelized. We also present a method to exploit symmetry to reduce table sizes, samples needed, and runtime during the reduction phase. The results apply to two-power cyclotomic Ring-LWE with parameters proposed for practical use (including all splitting types).
{"title":"Algebraic aspects of solving Ring-LWE, including ring-based improvements in the Blum-Kalai-Wasserman algorithm","authors":"Katherine E. Stange","doi":"10.1137/19M1280442","DOIUrl":"https://doi.org/10.1137/19M1280442","url":null,"abstract":"We provide a reduction of the Ring-LWE problem to Ring-LWE problems in subrings, in the presence of samples of a restricted form (i.e. $(a,b)$ such that $a$ is restricted to a multiplicative coset of the subring). To create and exploit such restricted samples, we propose Ring-BKW, a version of the Blum-Kalai-Wasserman algorithm which respects the ring structure. Off-the-shelf BKW dimension reduction (including coded-BKW and sieving) can be used for the reduction phase. Its primary advantage is that there is no need for back-substitution, and the solving/hypothesis-testing phase can be parallelized. We also present a method to exploit symmetry to reduce table sizes, samples needed, and runtime during the reduction phase. The results apply to two-power cyclotomic Ring-LWE with parameters proposed for practical use (including all splitting types).","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"11 1","pages":"366-387"},"PeriodicalIF":1.2,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74462946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded K [ x, y ]-module M , where K is a field. The algorithm takes as input a short chain complex of free modules X f −→ Y g −→ Z such that M ∼ = ker g/ im f . It runs in time O ( | X | 3 + | Y | 3 + | Z | 3 ) and requires O ( | X | 2 + | Y | 2 + | Z | 2 ) memory, where | · | denotes the rank. Given the presentation computed by our algorithm, the bigraded Betti numbers of M are readily computed. Our approach is based on a simple matrix reduction algorithm, slight variants of which compute kernels of morphisms between free modules, minimal generating sets, and Gr¨obner bases. Our algorithm for computing minimal presentations has been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology. In experiments on topological data analysis problems, our implementation outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin.
受拓扑数据分析应用的启发，我们给出了一种有效的算法来计算K [x, y]-模块M的(最小)表示，其中K是一个字段。该算法以自由模X f−→Y g−→Z的短链复合体作为输入，使得M ~ = ker g/ im f。它的运行时间为O (| X | 3 + | Y | 3 + | Z | 3)，并且需要O (| X | 2 + | Y | 2 + | Z | 2)内存，其中|·|表示rank。给出了该算法计算的表示形式，可以很容易地计算出M的分级贝蒂数。我们的方法是基于一个简单的矩阵约简算法，它可以计算自由模块、最小生成集和Gr¨obner基之间的态射核。我们计算最小表示的算法已经在RIVET中实现，RIVET是一个用于可视化和分析双参数持久同源的软件工具。在拓扑数据分析问题的实验中，我们的实现大大优于标准的计算交换代数包Singular和Macaulay2。
{"title":"Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology","authors":"M. Lesnick, Matthew L. Wright","doi":"10.1137/20m1388425","DOIUrl":"https://doi.org/10.1137/20m1388425","url":null,"abstract":"Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded K [ x, y ]-module M , where K is a field. The algorithm takes as input a short chain complex of free modules X f −→ Y g −→ Z such that M ∼ = ker g/ im f . It runs in time O ( | X | 3 + | Y | 3 + | Z | 3 ) and requires O ( | X | 2 + | Y | 2 + | Z | 2 ) memory, where | · | denotes the rank. Given the presentation computed by our algorithm, the bigraded Betti numbers of M are readily computed. Our approach is based on a simple matrix reduction algorithm, slight variants of which compute kernels of morphisms between free modules, minimal generating sets, and Gr¨obner bases. Our algorithm for computing minimal presentations has been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology. In experiments on topological data analysis problems, our implementation outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"70 1","pages":"267-298"},"PeriodicalIF":1.2,"publicationDate":"2019-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87186093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that one can reconstruct the shape of a room with planar walls from the first-order echoes received by four non-planar microphones placed on a drone with generic position and orientation. Both the cases where the source is located in the room and on the drone are considered. If the microphone positions are picked at random, then with probability one, the location of any wall is correctly reconstructed as long as it is heard by four microphones. Our algorithm uses a simple echo sorting criterion to recover the wall assignments for the echoes. We prove that, if the position and orientation of the drone on which the microphones are mounted do not lie on a certain set of dimension at most 5 in the 6-dimensional space of all drone positions and orientations, then the wall assignment obtained through our echo sorting criterion must be the right one and thus the reconstruction obtained through our algorithm is correct. Our proof uses methods from computational commutative algebra.
{"title":"A Drone Can Hear the Shape of a Room","authors":"M. Boutin, G. Kemper","doi":"10.1137/19m1248534","DOIUrl":"https://doi.org/10.1137/19m1248534","url":null,"abstract":"We show that one can reconstruct the shape of a room with planar walls from the first-order echoes received by four non-planar microphones placed on a drone with generic position and orientation. Both the cases where the source is located in the room and on the drone are considered. If the microphone positions are picked at random, then with probability one, the location of any wall is correctly reconstructed as long as it is heard by four microphones. Our algorithm uses a simple echo sorting criterion to recover the wall assignments for the echoes. We prove that, if the position and orientation of the drone on which the microphones are mounted do not lie on a certain set of dimension at most 5 in the 6-dimensional space of all drone positions and orientations, then the wall assignment obtained through our echo sorting criterion must be the right one and thus the reconstruction obtained through our algorithm is correct. Our proof uses methods from computational commutative algebra.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"53 1","pages":"123-140"},"PeriodicalIF":1.2,"publicationDate":"2019-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76499015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01Epub Date: 2019-03-14DOI: 10.1137/18m1194134
Elizabeth S Allman, Colby Long, John A Rhodes
The log-det distance between two aligned DNA sequences was introduced as a tool for statistically consistent inference of a gene tree under simple non-mixture models of sequence evolution. Here we prove that the log-det distance, coupled with a distance-based tree construction method, also permits consistent inference of species trees under mixture models appropriate to aligned genomic-scale sequences data. Data may include sites from many genetic loci, which evolved on different gene trees due to incomplete lineage sorting on an ultrametric species tree, with different time-reversible substitution processes. The simplicity and speed of distance-based inference suggests log-det based methods should serve as benchmarks for judging more elaborate and computationally-intensive species trees inference methods.
两个对齐的 DNA 序列之间的对数距离（log-det distance）被引入作为一种工具，用于在简单的非混合物序列进化模型下对基因树进行统计上一致的推断。在这里，我们证明对数-det 距离与基于距离的树构建方法相结合，也能在适合基因组尺度序列数据的混合模型下一致地推断物种树。数据可能包括来自许多基因位点的位点，这些位点在不同的基因树上进化，这是因为超对称物种树上的世系排序不完全，具有不同的时间可逆替换过程。基于距离推断的简单性和速度表明，基于 log-det 的方法应作为判断更复杂和计算密集型物种树推断方法的基准。
{"title":"SPECIES TREE INFERENCE FROM GENOMIC SEQUENCES USING THE LOG-DET DISTANCE.","authors":"Elizabeth S Allman, Colby Long, John A Rhodes","doi":"10.1137/18m1194134","DOIUrl":"10.1137/18m1194134","url":null,"abstract":"<p><p>The log-det distance between two aligned DNA sequences was introduced as a tool for statistically consistent inference of a gene tree under simple non-mixture models of sequence evolution. Here we prove that the log-det distance, coupled with a distance-based tree construction method, also permits consistent inference of species trees under mixture models appropriate to aligned genomic-scale sequences data. Data may include sites from many genetic loci, which evolved on different gene trees due to incomplete lineage sorting on an ultrametric species tree, with different time-reversible substitution processes. The simplicity and speed of distance-based inference suggests log-det based methods should serve as benchmarks for judging more elaborate and computationally-intensive species trees inference methods.</p>","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"3 1","pages":"107-127"},"PeriodicalIF":1.2,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7643864/pdf/nihms-1554294.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"38579242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We determine the optimal orthogonal matrices $R in {{O}}(n)$ which minimize the symmetrized Euclidean distance $Wcolon {{O}}(n) to Bbb{R}, ; W(R,;D) ;:=; vertvert{{sym}(R D - mathbbm{1})...
我们确定了最优正交矩阵$R in {{O}}(n)$，使对称欧几里得距离$Wcolon {{O}}(n) 到Bbb{R}， ;W (R ; D) ;: = ;vertvert{{sym}(R D - mathbbm{1})…
{"title":"Explicit Global Minimization of the Symmetrized Euclidean Distance by a Characterization of Real Matrices with Symmetric Square","authors":"P. Neff, Andreas Fischle, L. Borisov","doi":"10.1137/18M1179663","DOIUrl":"https://doi.org/10.1137/18M1179663","url":null,"abstract":"We determine the optimal orthogonal matrices $R in {{O}}(n)$ which minimize the symmetrized Euclidean distance $Wcolon {{O}}(n) to Bbb{R}, ; W(R,;D) ;:=; vertvert{{sym}(R D - mathbbm{1})...","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"197 1","pages":"31-43"},"PeriodicalIF":1.2,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89300394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We describe a new approach to certifying the global nonnegativity of multivariate polynomials by solving hyperbolic optimization problems---a class of convex optimization problems that generalize semidefinite programs. We show how to produce families of nonnegative polynomials (which we call hyperbolic certificates of nonnegativity) from any hyperbolic polynomial. We investigate the pairs $(n,d)$ for which there is a hyperbolic polynomial of degree $d$ in $n$ variables such that an associated hyperbolic certificate of nonnegativity is not a sum of squares. If $dgeq 4$ we show that this occurs whenever $ngeq 4$. In the degree three case, we find an explicit hyperbolic cubic in $43$ variables that gives hyperbolic certificates that are not sums of squares. As a corollary, we obtain the first known hyperbolic cubic no power of which has a definite determinantal representation. Our approach also allows us to show that, given a cubic $p$, and a direction $e$, the decision problem "Is $p$ hyperbolic with respect to $e$?" is co-NP hard.
{"title":"Certifying Polynomial Nonnegativity via Hyperbolic Optimization","authors":"J. Saunderson","doi":"10.1137/19m1253551","DOIUrl":"https://doi.org/10.1137/19m1253551","url":null,"abstract":"We describe a new approach to certifying the global nonnegativity of multivariate polynomials by solving hyperbolic optimization problems---a class of convex optimization problems that generalize semidefinite programs. We show how to produce families of nonnegative polynomials (which we call hyperbolic certificates of nonnegativity) from any hyperbolic polynomial. We investigate the pairs $(n,d)$ for which there is a hyperbolic polynomial of degree $d$ in $n$ variables such that an associated hyperbolic certificate of nonnegativity is not a sum of squares. If $dgeq 4$ we show that this occurs whenever $ngeq 4$. In the degree three case, we find an explicit hyperbolic cubic in $43$ variables that gives hyperbolic certificates that are not sums of squares. As a corollary, we obtain the first known hyperbolic cubic no power of which has a definite determinantal representation. Our approach also allows us to show that, given a cubic $p$, and a direction $e$, the decision problem \"Is $p$ hyperbolic with respect to $e$?\" is co-NP hard.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"5 1","pages":"661-690"},"PeriodicalIF":1.2,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89611214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
There is an error in Theorem 3.1 of [S. Shankar and P. Rocha, SIAM J. Appl. Algebra Geometry, 2 (2018), pp. 410--427], which propagates to the statements of Theorems 3.2, 4.1, and 4.2. The correct ...
{"title":"Corrigendum: The Generic Degree of Autonomy","authors":"S. Shankar","doi":"10.1137/18M1210253","DOIUrl":"https://doi.org/10.1137/18M1210253","url":null,"abstract":"There is an error in Theorem 3.1 of [S. Shankar and P. Rocha, SIAM J. Appl. Algebra Geometry, 2 (2018), pp. 410--427], which propagates to the statements of Theorems 3.2, 4.1, and 4.2. The correct ...","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"52 1","pages":"172-174"},"PeriodicalIF":1.2,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80532422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Some of the most common mathematical models in biology, chemistry, physics, and engineering are polynomial dynamical systems, i.e., systems of differential equations with polynomial right-hand side...
{"title":"Polynomial Dynamical Systems, Reaction Networks, and Toric Differential Inclusions","authors":"G. Craciun","doi":"10.1137/17M1129076","DOIUrl":"https://doi.org/10.1137/17M1129076","url":null,"abstract":"Some of the most common mathematical models in biology, chemistry, physics, and engineering are polynomial dynamical systems, i.e., systems of differential equations with polynomial right-hand side...","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"1 1","pages":"87-106"},"PeriodicalIF":1.2,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73935735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we adapt the differential signature construction to the equivalence problem for complex plane algebraic curves under the actions of the projective group and its subgroups. Given an action of a group $G$, a signature map assigns to a plane algebraic curve another plane algebraic curve (a signature curve) in such a way that two generic curves have the same signatures if and only if they are $G$-equivalent. We prove that for any $G$-action, there exists a pair of rational differential invariants, called classifying invariants, that can be used to construct signatures. We derive a formula for the degree of a signature curve in terms of the degree of the original curve, the size of its symmetry group and some quantities depending on a choice of classifying invariants. For the full projective group, as well as for its affine, special affine and special Euclidean subgroups, we give explicit sets of rational classifying invariants and derive a formula for the degree of the signature curve of a generic curve as a quadratic function of the degree of the original curve. We show that this generic degree is the sharp upper bound.
{"title":"Differential Signatures of Algebraic Curves","authors":"I. Kogan, Michael Ruddy, C. Vinzant","doi":"10.1137/19m1242859","DOIUrl":"https://doi.org/10.1137/19m1242859","url":null,"abstract":"In this paper, we adapt the differential signature construction to the equivalence problem for complex plane algebraic curves under the actions of the projective group and its subgroups. Given an action of a group $G$, a signature map assigns to a plane algebraic curve another plane algebraic curve (a signature curve) in such a way that two generic curves have the same signatures if and only if they are $G$-equivalent. We prove that for any $G$-action, there exists a pair of rational differential invariants, called classifying invariants, that can be used to construct signatures. We derive a formula for the degree of a signature curve in terms of the degree of the original curve, the size of its symmetry group and some quantities depending on a choice of classifying invariants. For the full projective group, as well as for its affine, special affine and special Euclidean subgroups, we give explicit sets of rational classifying invariants and derive a formula for the degree of the signature curve of a generic curve as a quadratic function of the degree of the original curve. We show that this generic degree is the sharp upper bound.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"59 1","pages":"185-226"},"PeriodicalIF":1.2,"publicationDate":"2018-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81960451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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