Exactness in numerical algebraic computations
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1 Exactness in numerical algebraic computations Dan Bates Jon Hauenstein Tim McCoy Chris Peterson Andrew Sommese Wednesday, December 17, 2008 MSRI Workshop on Algebraic Statstics
2 Main goals for this talk 2/33
3 Main goals for this talk 1. Give a quick and dirty primer on the basics of numerical algebraic geometry (for newcomers). 2/33
4 Main goals for this talk 1. Give a quick and dirty primer on the basics of numerical algebraic geometry (for newcomers). 2. Report on the (early) progress on a new project in this direction, perhaps entirely unrelated to algebraic statistics. 2/33
5 The Gameplan Motivation Methods for recovering exactness Available numerical data The new algorithm An example Unresolved issues 3/33
6 Motivation (exactness) 4/33
7 Motivation (exactness) We want to study polynomial systems in Z[x 1,,x n ] and their solution sets in C n. 4/33
8 Motivation (exactness) We want to study polynomial systems in Z[x 1,,x n ] and their solution sets in C n. There are symbolic methods, which have benefits and drawbacks. 4/33
9 Motivation (exactness) We want to study polynomial systems in Z[x 1,,x n ] and their solution sets in C n. There are symbolic methods, which have benefits and drawbacks. There are also numerical methods, also with benefits and drawbacks. 4/33
10 Motivation (exactness) We want to study polynomial systems in Z[x 1,,x n ] and their solution sets in C n. There are symbolic methods, which have benefits and drawbacks. There are also numerical methods, also with benefits and drawbacks. Today, we ll look at a bridge from numerical computation to symbolic information. 4/33
11 Motivation (exactness) We want to study polynomial systems in Z[x 1,,x n ] and their solution sets in C n. There are symbolic methods, which have benefits and drawbacks. There are also numerical methods, also with benefits and drawbacks. Today, we ll look at a bridge from numerical computation to symbolic information. WARNING: This project may not have anything to do with algebraic statistics (?). 4/33
12 Motivation (exactness) More specifically: Given some approximations of points on each component, can we: (a) find a polynomial (with exact coefficients) that vanishes on a given irreducible component? (b) find an ideal for each irreducible component? (c) recover the original ideal? 5/33
13 Motivation (exactness) More specifically: Given some approximations of points on each component, can we: (a) find a polynomial (with exact coefficients) that vanishes on a given irreducible component? (b) find an ideal for each irreducible component? (c) recover the original ideal? Who cares anyway? Numerical methods may be more efficient, at least in certain cases. 5/33
14 Motivation (NAG) 6/33
15 Motivation (NAG) Basic homotopy continuation: 6/33
16 Motivation (NAG) Basic homotopy continuation: Want the isolated (complex) solutions of the (complex) target system F(z)=0. 6/33
17 Motivation (NAG) Basic homotopy continuation: Want the isolated (complex) solutions of the (complex) target system F(z)=0. Choose a start system G(z). 6/33
18 Motivation (NAG) Basic homotopy continuation: Want the isolated (complex) solutions of the (complex) target system F(z)=0. Choose a start system G(z). Form H(z,t) = F(z) (1-t) + G(z) t such that: - H(z,0) = F(z). - H(z,1) = G(z). 6/33
19 Motivation (NAG) Basic homotopy continuation: Want the isolated (complex) solutions of the (complex) target system F(z)=0. Choose a start system G(z). Form H(z,t) = F(z) (1-t) + G(z) t such that: - H(z,0) = F(z). - H(z,1) = G(z). (WARNING: As stated, this is incorrect.) 6/33
20 Motivation (NAG) Basic homotopy continuation: t=0 t=1 t 7/33
21 Motivation (NAG) Basic homotopy continuation: t=0 t=1 t 7/33
22 Motivation (NAG) Basic homotopy continuation: t=0 t=1 t 7/33
23 Motivation (NAG) Basic homotopy continuation: t=0 t=1 t 7/33
24 Motivation (NAG) Basic homotopy continuation: t=0 t=1 t Number of paths depends on choice of g(z): - At most the Bézout number, and - At least the BKK number. 7/33
25 Motivation (NAG) Basic homotopy continuation: t=0 t=1 t 7/33
26 Motivation (NAG) Basic homotopy continuation: t=0 t=1 t 7/33
27 Motivation (NAG) Basic homotopy continuation: t=0 t=1 t 7/33
28 Motivation (NAG) Basic homotopy continuation: t=0 t=1 t 7/33
29 Motivation (NAG) Basic homotopy continuation: t=0 t=1 t 7/33
30 Motivation (NAG) Basic homotopy continuation: t=0 t=1 t 7/33
31 Motivation (NAG) Basic homotopy continuation: t=0 t=1 t (Also not entirely correct.) 7/33
32 Motivation (NAG)... 8/33
33 Motivation (NAG)... (Just add linear equations to F(z).) 8/33
34 Motivation (NAG)... 8/33
35 Motivation (NAG)... (Remove one linear equation from F(z).) 8/33
36 Motivation (NAG)... 8/33
37 Motivation (NAG)... (Membership or LDT) 8/33
38 Motivation (NAG)... 8/33
39 Motivation (NAG)... 8/33
40 Motivation (NAG)... 8/33
41 Motivation (NAG)... 8/33
42 Motivation (NAG) Bottom line: If you need to find solutions of polynomial systems, there are numerical methods for doing this. 9/33
43 Motivation (NAG) Bottom line: If you need to find solutions of polynomial systems, there are numerical methods for doing this. In particular, these methods work for large systems. 9/33
44 Motivation (NAG) Bottom line: If you need to find solutions of polynomial systems, there are numerical methods for doing this. In particular, these methods work for large systems. For more on numerical algebraic geometry, see: - the 2005 book by Sommese-Wampler - me - Anton Leykin - Philipp Rostalski - (others) 9/33
45 Motivation (NAG) A few more advanced methods that may interest you: - Component sampling and membership - Parameter continuation (for parameterized systems) - Regularity and multiplicity structure of a zero-scheme (Dayton-Zeng 04 or B-Peterson-Sommese 05) - Local dimension test (B-Peterson-Sommese 08) - Numerical primary decomposition (Leykin 08) 10/33
46 Motivation (NAG) Software: - PHCPack (Jan Verschelde) - Bertini (B, Hauenstein, Sommese, Charles Wampler) released last week - Parallel and Mac versions available now - HOM4PS mixedvol (T.Y. Li,.) Others: - HomLab (Wampler) - PHoM (Kojima,.) - HOMPACK (Layne Watson,.) 11/33
47 Motivation (NAG) Software: - PHCPack (Jan Verschelde) - Bertini (B, Hauenstein, Sommese, Charles Wampler) released last week - Parallel and Mac versions available now - HOM4PS mixedvol (T.Y. Li,.) Others: - HomLab (Wampler) - PHoM (Kojima,.) - HOMPACK (Layne Watson,.) 11/33
48 Motivation (NAG) A word about words: 12/33
49 Motivation (NAG) A word about words: Numerical algebraic geometry: Numerical methods for algebraic geometry based on homotopy continuation. 12/33
50 Motivation (NAG) A word about words: Numerical algebraic geometry: Numerical methods for algebraic geometry based on homotopy continuation. This is probably too restrictive: - Stetter (Numerical polynomial algebra) - ApCoCoA (APCOA) - Kaltofen/Szanto (clusters of roots) - Shub/Smale (clusters of roots) - others 12/33
51 Motivation (exactness) Back to recovery of exactness (finally): We want functions. Why not interpolate? 13/33
52 Motivation (exactness) Back to recovery of exactness (finally): We want functions. Why not interpolate? Example: Twisted cubic (degree 3 curve) I = <xy-z, y 2 -xz, x 2 -y>. 13/33
53 Motivation (exactness) Back to recovery of exactness (finally): We want functions. Why not interpolate? Example: Twisted cubic (degree 3 curve) I = <xy-z, y 2 -xz, x 2 -y>. 20 (exact) sample points: 3 generators, 7 simple elements of I. 13/33
54 Motivation (exactness) Back to recovery of exactness (finally): We want functions. Why not interpolate? Example: Twisted cubic (degree 3 curve) I = <xy-z, y 2 -xz, x 2 -y>. 20 (exact) sample points: 3 generators, 7 simple elements of I. Added perturbations to sample points: Numerical results (no nonzero coefficients). 13/33
55 Motivation (exactness) Problems with interpolation: - Numerical instability. - Yields inexact coefficients, so worthless for us anyway. 14/33
56 Motivation (exactness) Problems with interpolation: - Numerical instability. - Yields inexact coefficients, so worthless for us anyway. Q: So how do we move from real numbers to rational numbers (or integers)? 14/33
57 Motivation (exactness) Problems with interpolation: - Numerical instability. - Yields inexact coefficients, so worthless for us anyway. Q: So how do we move from real numbers to rational numbers (or integers)? A: Lean on some algorithms related to multivariate factorization, the numerical approximation of numerical constants, etc. 14/33
58 The Gameplan Motivation Methods for recovering exactness Available numerical data The new algorithm An example Unresolved issues
59 Methods for Recovering Exactness Given floating point numbers x1,, xn, how can we find integers a1,, an such that a1x1+ +anxn 0. 15/33
60 Methods for Recovering Exactness Given floating point numbers x1,, xn, how can we find integers a1,, an such that a1x1+ +anxn 0. All methods boil down to some numerical linear algebra. 15/33
61 Methods for Recovering Exactness Given floating point numbers x1,, xn, how can we find integers a1,, an such that a1x1+ +anxn 0. All methods boil down to some numerical linear algebra. For this project, we treat these as blackboxes. 15/33
62 Methods for Recovering Exactness Given floating point numbers x1,, xn, how can we find integers a1,, an such that a1x1+ +anxn 0. All methods boil down to some numerical linear algebra. For this project, we treat these as blackboxes. 1. LLL (Lenstra-Lenstra-Lovasz, 82) Canonical choice. 15/33
63 Methods for Recovering Exactness Given floating point numbers x1,, xn, how can we find integers a1,, an such that a1x1+ +anxn 0. All methods boil down to some numerical linear algebra. For this project, we treat these as blackboxes. 1. LLL (Lenstra-Lenstra-Lovasz, 82) Canonical choice. Original motivation: Find a nearly-orthogonal basis which contains a short vector ( shortest is NP-hard). 15/33
64 Methods for Recovering Exactness Main application: Factorization of polynomials: Univariate: Lenstra-Lenstra-Lovasz, 1982 Dense multivariate: Kaltofen, 1985 Sparse multivariate: von zur Gathen-Kaltofen, 1985 Using straight-line programs: Kaltofen, 1989 Using blackbox representation: Kaltofen-Trager, /33
65 Methods for Recovering Exactness Some drawbacks: n Short isn t very short. ( 2 f f ) It is a precision hog. There is no guarantee that failure to find a short vector means that there is no relation (could be precision). It is numerically unstable. The time complexity is higher than with other methods. 17/33
66 Methods for Recovering Exactness Some drawbacks: n Short isn t very short. ( 2 f f ) It is a precision hog. There is no guarantee that failure to find a short vector means that there is no relation (could be precision). It is numerically unstable. The time complexity is higher than with other methods. Q: Why is this the standard? 17/33
67 Methods for Recovering Exactness Some drawbacks: n Short isn t very short. ( 2 f f ) It is a precision hog. There is no guarantee that failure to find a short vector means that there is no relation (could be precision). It is numerically unstable. The time complexity is higher than with other methods. Q: Why is this the standard? A: It was the first polynomial-time algorithm. 17/33
68 Methods for Recovering Exactness 2. Ferguson-Forcade (1979) First algorithm for more than two real numbers. Horrific complexity, numerically unstable, etc. 3. PSOS (Ferguson, 1988) 4. HJLS (Ferguson-Bailey, 1992) 18/33
69 Methods for Recovering Exactness 5. PSLQ (Ferguson-Bailey, 92, 99 (with Arno)) Original motivation: Given a set of floating point numbers x1,, xn, find integers a1,, an such that a1x1+ +anxn 0. 19/33
70 Methods for Recovering Exactness 5. PSLQ (Ferguson-Bailey, 92, 99 (with Arno)) Original motivation: Given a set of floating point numbers x1,, xn, find integers a1,, an such that a1x1+ +anxn 0. Used for the numerical approximation of constants such as π(see Bailey, Borwein, & Plouffe, 97). 19/33
71 Methods for Recovering Exactness Nice properties: Better run times than the others. Requires very little working precision. Clear signal when precision is the problem (so it yields proof that there are no relations). Experimentally, it seems to be clear when an integer relation is not good (based on the number of digits in the coefficients). 20/33
72 Methods for Recovering Exactness Implementations: There are several standalone + Maple + SAGE, etc. We use Maple for now (for better or for worse). Homemade versions are easy the recipe is short and clear. 21/33
73 The Gameplan Motivation Methods for recovering exactness Available numerical data (back to the project) The new algorithm An example Unresolved issues
74 Available Numerical Data In numerical algebraic geometry, we use numerical (homotopy) methods to compute witness sets. Important note: Witness points can be computed (easily) to extremely high precision thousands of digits. 22/33
75 Available Numerical Data In numerical algebraic geometry, we use numerical (homotopy) methods to compute witness sets. Important note: Witness points can be computed (easily) to extremely high precision thousands of digits. Example: Twisted cubic, C. 22/33
76 Available Numerical Data In numerical algebraic geometry, we use numerical (homotopy) methods to compute witness sets. Important note: Witness points can be computed (easily) to extremely high precision thousands of digits. Example: Twisted cubic, C. We find deg(c) = 3 generic points on C, to any level of precision: 100 digits 22/33
77 Available Numerical Data In numerical algebraic geometry, we use numerical (homotopy) methods to compute witness sets. Important note: Witness points can be computed (easily) to extremely high precision thousands of digits. Example: Twisted cubic, C. We find deg(c) = 3 generic points on C, to any level of precision: 100 digits x= e *i 22/33
78 Available Numerical Data In numerical algebraic geometry, we use numerical (homotopy) methods to compute witness sets. Important note: Witness points can be computed (easily) to extremely high precision thousands of digits. Example: Twisted cubic, C. We find deg(c) = 3 generic points on C, to any level of precision: 200 digits x= e e-2*i
79 Available Numerical Data In numerical algebraic geometry, we use numerical (homotopy) methods to compute witness sets. Important note: Witness points can be computed (easily) to extremely high precision thousands of digits. Example: Twisted cubic, C. We find deg(c) = 3 generic points on C, to any level of precision: 1000 digits X= e e0*i
80 Available Numerical Data X= In numerical algebraic geometry, we use numerical (homotopy) methods to compute witness sets Important note: Witness points can be computed (easily) to extremely high precision thousands of digits e Example: Twisted cubic, C We find deg(c) = 3 generic points on C, to any level of precision: digits e0*i
81 The Gameplan Motivation Methods for recovering exactness Available numerical data The new algorithm An example Unresolved issues 23/33
82 The new algorithm Input: Point P on some reduced*, irreducible component C of some complex algebraic set, available to any level of precision. Output: A set of generators for the ideal defining C. *: We have an algorithm for the nonreduced case, too, though that is for another talk. 24/33
83 The new algorithm For D (degree) from 1 to MAX_D: 25/33
84 The new algorithm For D (degree) from 1 to MAX_D: PREC = MIN_PREC. Form the degree D Veronese embedding P D of point P. 25/33
85 The new algorithm For D (degree) from 1 to MAX_D: PREC = MIN_PREC. Form the degree D Veronese embedding P D of point P. While not done: (loop within this degree) 25/33
86 The new algorithm For D (degree) from 1 to MAX_D: PREC = MIN_PREC. Form the degree D Veronese embedding P D of point P. While not done: (loop within this degree) Run PSLQ to obtain polynomial f or detect lack of precision. 25/33
87 The new algorithm For D (degree) from 1 to MAX_D: PREC = MIN_PREC. Form the degree D Veronese embedding P D of point P. While not done: (loop within this degree) Run PSLQ to obtain polynomial f or detect lack of precision. If more precision is needed, increase PREC and restart loop. 25/33
88 The new algorithm For D (degree) from 1 to MAX_D: PREC = MIN_PREC. Form the degree D Veronese embedding P D of point P. While not done: (loop within this degree) Run PSLQ to obtain polynomial f or detect lack of precision. If more precision is needed, increase PREC and restart loop. Otherwise, decide if f is actually a relation and if f is new. 25/33
89 The new algorithm For D (degree) from 1 to MAX_D: PREC = MIN_PREC. Form the degree D Veronese embedding P D of point P. While not done: (loop within this degree) Run PSLQ to obtain polynomial f or detect lack of precision. If more precision is needed, increase PREC and restart loop. Otherwise, decide if f is actually a relation and if f is new. If f is a new relation, store it, trim PD, and restart loop. 25/33
90 The new algorithm For D (degree) from 1 to MAX_D: PREC = MIN_PREC. Form the degree D Veronese embedding P D of point P. While not done: (loop within this degree) Run PSLQ to obtain polynomial f or detect lack of precision. If more precision is needed, increase PREC and restart loop. Otherwise, decide if f is actually a relation and if f is new. If f is a new relation, store it, trim PD, and restart loop. If f is not new, just trim PD and restart loop. 25/33
91 The new algorithm For D (degree) from 1 to MAX_D: PREC = MIN_PREC. Form the degree D Veronese embedding P D of point P. While not done: (loop within this degree) Run PSLQ to obtain polynomial f or detect lack of precision. If more precision is needed, increase PREC and restart loop. Otherwise, decide if f is actually a relation and if f is new. If f is a new relation, store it, trim PD, and restart loop. If f is not new, just trim PD and restart loop. If f is not a relation (i.e., it is junk), break to outer loop. 25/33
92 The new algorithm MAX_D: Castelnuovo-Mumford regularity. (better??) actually a relation: The number of digits in meaningless integer relations is around PREC/vector size. Can we make a mistake? Yes, but we can correct it. f is new : Any p(x)f(x) where f(x) is a generator will also be in the ideal. So, we see if any potentially new generator is in the ideal formed by the previously discovered generators. 26/33
93 The new algorithm Q: Will this algorithm terminate? 27/33
94 The new algorithm Q: Will this algorithm terminate? A: Yes. The degree of the generators (in a minimal set of generators) is bounded. We could even compute the bound. 27/33
95 The new algorithm Q: Will this algorithm terminate? A: Yes. The degree of the generators (in a minimal set of generators) is bounded. We could even compute the bound. Q: Can we detect whether we are done? 27/33
96 The new algorithm Q: Will this algorithm terminate? A: Yes. The degree of the generators (in a minimal set of generators) is bounded. We could even compute the bound. Q: Can we detect whether we are done? A: Yes, numerically and/or symbolically. This is good since the bounds are awful. 27/33
97 The Gameplan Motivation Methods for recovering exactness Available numerical data The new algorithm An example Unresolved issues
98 An example To make a long story short: Example: Twisted cubic. Using 100 digits (maybe less?), recovered all 3 generators: 28/33
99 An example To make a long story short: Example: Twisted cubic. Using 100 digits (maybe less?), recovered all 3 generators: 28/33
100 An example To make a long story short: Example: Twisted cubic. Using 100 digits (maybe less?), recovered all 3 generators: 28/33
101
102 30/33
103 The Gameplan Motivation Methods for recovering exactness Available numerical data The new algorithm An example Unresolved issues 31/33
104 Unresolved issues Detect how much precision we will need a priori (either in the data or for computation)? Are there better bounds on the degrees of the generators? Complexity? Is this really faster than prime decomposition? Stability? I.e., suppose we know our points to 3 digits. Could we accidentally pick out some other algebraic set? Avoid Gröbner bases, control growth in Veronese embedding? 32/33
105 What I hope you get out of this: You can find isolated solutions of polynomial systems with numerical methods. 33/33
106 What I hope you get out of this: You can find isolated solutions of polynomial systems with numerical methods. You can detect positive-dimensional components numerically by finding points on them. 33/33
107 What I hope you get out of this: You can find isolated solutions of polynomial systems with numerical methods. You can detect positive-dimensional components numerically by finding points on them. There is a way to move from numerical data to exact information, in various settings. 33/33
108 What I hope you get out of this: You can find isolated solutions of polynomial systems with numerical methods. You can detect positive-dimensional components numerically by finding points on them. There is a way to move from numerical data to exact information, in various settings. New approach for the salmon conjecture? Can you find a point on your algebraic set? 33/33
109 What I hope you get out of this: You can find isolated solutions of polynomial systems with numerical methods. You can detect positive-dimensional components numerically by finding points on them. There is a way to move from numerical data to exact information, in various settings. New approach for the salmon conjecture? Can you find a point on your algebraic set? Max Buot: Maximum Likelihood Equations. 33/33
110 What I hope you get out of this: You can find isolated solutions of polynomial systems with numerical methods. You can detect positive-dimensional components numerically by finding points on them. There is a way to move from numerical data to exact information, in various settings. New approach for the salmon conjecture? Can you find a point on your algebraic set? Max Buot: Maximum Likelihood Equations. I will bet that Bernd has some ideas. 33/33
111 A few references A. Lenstra, H. Lenstra, & L. Lovasz. Factoring polynomials with rational coefficients. Math. Ann. 261 (1982), (LLL) D. Bailey & H. Ferguson. A polynomial time, numerically stable integer relation algorithm. SRC Technical Report SRC-TR (1981), (PSLQ) J. von zur Gathen & J. Gerhard. Modern Computer Algebra. Cambridge (2003). (general lattice basis reduction) A. Sommese & C. Wampler. The Numerical Solution of Systems of Polynomials: Arising in Engineering And Science. World Scientific (2006). (numerical algebraic geometry) Thank you!
112 Methods for Recovering Exactness Example (from von zur Gathen-Gerhard): Lattice generated by (12,2), (13,4). LLL: (1,2), (9,-4). How short is short? This time, shortest. Often nearly the shortest. Proven bound: f n f
113 Methods for Recovering Exactness Basic procedure: Gram-Schmidt, with a twist. j 1 < v j, ei > Usual Gram Schmidt: e j = v j e i= 1 < e, e > j i i LLL s Gram Schmidt : e j j 1 v j, e i = v j i=1 < e j, ei < v > > e i LLL uses this idea, along with reordering the vectors between steps (to work towards short vectors). Result: Basis vectors grow by a factor of 2 (or more).
114 Methods for Recovering Exactness Main idea: Replace Gram-Schmidt with Householder transformations for numerical stability (in addition to various other changes). Nice properties: Nice properties: Better run times than the others. Requires very little working precision. Clear signal when precision is the problem (so it yields proof that there are no relations). Experimentally, it seems to be clear when an integer relation is not good (based on the number of digits in the coefficients).
115 Methods for Recovering Exactness Popular Application: (Bailey-Borwein-Plouffe, 97) Apply PSLQ to (X1, X2,, X8, π) where Xj = = k 0 16 k 1 (8k + j) = 1 j (8 + j) (8 2 + j) PSLQ yields (4,0,0,-2,-1,-1,0,0,-1). So π = Pi = i = 0 i= i + 1 8i + 4 8i + 5 8i + 6 i Pi 1 16, so we get 1 hexadecimal digit per step. P i
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