Permanent estimators via random matrices Mark Rudelson joint work - - PowerPoint PPT Presentation

Permanent estimators via random matrices

Mark Rudelson

joint work with Ofer Zeitouni

Department of Mathematics University of Michigan

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Permanent of a matrix

Let A be an n × n matrix with ai,j ≥ 0.

Permanent of A:

perm(A) =

• π∈Πn

n

• j=1

aj,π(j).

Mark Rudelson (Michigan) Permanent estimators via random matrices 2 / 25

Permanent of a matrix

Let A be an n × n matrix with ai,j ≥ 0.

Permanent of A:

perm(A) =

• π∈Πn

n

• j=1

aj,π(j).

Determinant of A:

det(A) =

• π∈Πn

sign(π)

n

• j=1

aj,π(j).

Mark Rudelson (Michigan) Permanent estimators via random matrices 2 / 25

Permanent of a matrix

Let A be an n × n matrix with ai,j ≥ 0.

Permanent of A:

perm(A) =

• π∈Πn

n

• j=1

aj,π(j).

Determinant of A:

det(A) =

• π∈Πn

sign(π)

n

• j=1

aj,π(j). Evaluation of determinants is fast: use e.g., triangularization by Gaussian elimination. Kalfoten–Villard algorithm: Running time: O(n2.7).

Mark Rudelson (Michigan) Permanent estimators via random matrices 2 / 25

Permanent of a matrix

Let A be an n × n matrix with ai,j ≥ 0.

Permanent of A:

perm(A) =

• π∈Πn

n

• j=1

aj,π(j). Evaluation of permanents is #P-complete (Valiant 1979) if there exists a polynomial-time algorithm for permanent evaluation, then any #P problem can be solved in polynomial time. Fast computation ⇒ P=NP.

Determinant of A:

det(A) =

• π∈Πn

sign(π)

n

• j=1

aj,π(j). Evaluation of determinants is fast: use e.g., triangularization by Gaussian elimination. Kalfoten–Villard algorithm: Running time: O(n2.7).

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Applications of permanents

Perfect matchings

Let Γ = (L, R, V) be an n × n bipartite graph.

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Applications of permanents

Perfect matchings

Let Γ = (L, R, V) be an n × n bipartite graph. A perfect matching is a bijection τ : E → R such that e → τ(e) for all e ∈ E.

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Applications of permanents

Perfect matchings

Let Γ = (L, R, V) be an n × n bipartite graph. A perfect matching is a bijection τ : E → R such that e → τ(e) for all e ∈ E.

Mark Rudelson (Michigan) Permanent estimators via random matrices 3 / 25

Applications of permanents

Perfect matchings

Let Γ = (L, R, V) be an n × n bipartite graph. A perfect matching is a bijection τ : E → R such that e → τ(e) for all e ∈ E. #(perfect matchings) = perm(A), where A is the adjacency matrix of the graph: ai,j = 1 if i → j.

Mark Rudelson (Michigan) Permanent estimators via random matrices 3 / 25

Deterministic bounds

Linial–Samorodnitsky–Wigderson algoritm: if perm(A) > 0, then one can find in polynomial time diagonal matrices D, D′ such that the renormalized matrix A′ = D′AD is almost doubly stochastic: 1 − ε <

n

• i=1

a′

i,j < 1 + ε,

for all j = 1, . . . , n 1 − ε <

n

• j=1

a′

i,j < 1 + ε,

for all i = 1, . . . , n

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Deterministic bounds

Linial–Samorodnitsky–Wigderson algoritm: if perm(A) > 0, then one can find in polynomial time diagonal matrices D, D′ such that the renormalized matrix A′ = D′AD is almost doubly stochastic: 1 − ε <

n

• i=1

a′

i,j < 1 + ε,

for all j = 1, . . . , n 1 − ε <

n

• j=1

a′

i,j < 1 + ε,

for all i = 1, . . . , n perm(A) = n

i=1 di · n j=1 d′ j · perm(A′)

Mark Rudelson (Michigan) Permanent estimators via random matrices 4 / 25

Deterministic bounds

Linial–Samorodnitsky–Wigderson algoritm: reduces permanent estimates to almost doubly stochastic matrices Van der Waerden conjecture, proved by Falikman and Egorychev: if A is doubly stochastic, then 1 ≥ perm(A) ≥ n! nn ≈ e−n Linial–Samorodnitsky–Wigderson algorithm estimates the permanent with the multiplicative error at most en

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Deterministic bounds

Linial–Samorodnitsky–Wigderson algoritm: reduces permanent estimates to almost doubly stochastic matrices Van der Waerden conjecture, proved by Falikman and Egorychev: if A is doubly stochastic, then 1 ≥ perm(A) ≥ n! nn ≈ e−n Linial–Samorodnitsky–Wigderson algorithm estimates the permanent with the multiplicative error at most en Bregman’s theorem (1973) implies that if A is doubly stochastic, and max ai,j ≤ t · min ai,j, then perm(A) ≤ e−n · nO(t2) Conclusion: if max ai,j ≤ t · min ai,j, then Linial–Samorodnitsky–Wigderson algoritm yields a multiplicative error nO(t2) Doesn’t cover matrices with zeros.

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Probabilistic estimates

Jerrum–Sinclair–Vigoda algorithm estimates the permanent of any matrix with constant multiplicative error with high probability.

Mark Rudelson (Michigan) Permanent estimators via random matrices 6 / 25

Probabilistic estimates

Jerrum–Sinclair–Vigoda algorithm estimates the permanent of any matrix with constant multiplicative error with high probability. Deficiency: running time is O(n10)

Mark Rudelson (Michigan) Permanent estimators via random matrices 6 / 25

Probabilistic estimates

Jerrum–Sinclair–Vigoda algorithm estimates the permanent of any matrix with constant multiplicative error with high probability. Deficiency: running time is O(n10) Godsil–Gutman estimator Let A1/2 be the matrix with entries a1/2

i,j .

Let R be an n × n random matrix with i.i.d. ±1 entries. Form the Hadamard product R ⊙ A1/2: wi,j = √ai,j · ri,j. Then perm(A) = E det2(R ⊙ A1/2). Estimator: perm(A) ≈ det2(R ⊙ A1/2).

Mark Rudelson (Michigan) Permanent estimators via random matrices 6 / 25

Probabilistic estimates

Jerrum–Sinclair–Vigoda algorithm estimates the permanent of any matrix with constant multiplicative error with high probability. Deficiency: running time is O(n10) Godsil–Gutman estimator Let A1/2 be the matrix with entries a1/2

i,j .

Let R be an n × n random matrix with i.i.d. ±1 entries. Form the Hadamard product R ⊙ A1/2: wi,j = √ai,j · ri,j. Then perm(A) = E det2(R ⊙ A1/2). Estimator: perm(A) ≈ det2(R ⊙ A1/2). Advantage: Godsil–Gutman estimator is faster than any other algorithm.

Mark Rudelson (Michigan) Permanent estimators via random matrices 6 / 25

Probabilistic estimates

Jerrum–Sinclair–Vigoda algorithm estimates the permanent of any matrix with constant multiplicative error with high probability. Deficiency: running time is O(n10) Godsil–Gutman estimator Let A1/2 be the matrix with entries a1/2

i,j .

Let R be an n × n random matrix with i.i.d. ±1 entries. Form the Hadamard product R ⊙ A1/2: wi,j = √ai,j · ri,j. Then perm(A) = E det2(R ⊙ A1/2). Estimator: perm(A) ≈ det2(R ⊙ A1/2). Advantage: Godsil–Gutman estimator is faster than any other algorithm. Deficiency: Godsil–Gutman estimator performs well for “generic” matrices, but fails for large classes of {0, 1} matrices, because of arithmetic issues.

Mark Rudelson (Michigan) Permanent estimators via random matrices 6 / 25

Barvinok’s estimator

Godsil–Gutman estimator Let A1/2 be the matrix with entries a1/2

i,j .

Let R be an n × n random matrix with i.i.d. ±1 entries. Form the Hadamard product R ⊙ A1/2. Then perm(A) = E det2(R ⊙ A1/2). Estimator: perm(A) ≈ det2(R ⊙ A1/2).

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Barvinok’s estimator

Barvinok’s estimator Let A1/2 be the matrix with entries a1/2

i,j .

Let G be an n × n random matrix with i.i.d. N(0, 1) entries. Form the Hadamard product G ⊙ A1/2. Then perm(A) = E det2(G ⊙ A1/2). Estimator: perm(A) ≈ det2(G ⊙ A1/2). Barvinok’s estimator has no arithmetic issues.

Mark Rudelson (Michigan) Permanent estimators via random matrices 7 / 25

Barvinok’s estimator

Barvinok’s estimator Let A1/2 be the matrix with entries a1/2

i,j .

Let G be an n × n random matrix with i.i.d. N(0, 1) entries. Form the Hadamard product G ⊙ A1/2. Then perm(A) = E det2(G ⊙ A1/2). Estimator: perm(A) ≈ det2(G ⊙ A1/2). Barvinok’s estimator has no arithmetic issues.

Theorem (Barvinok)

Let A be any n × n matrix. Then, with probability 1 − δ, ((1 − ε) · θ)n perm(A) ≤ det2(G ⊙ A1/2) ≤ C perm(A), where C is an absolute constant and θ = 0.28 for real Gaussian matrices; θ = 0.56 for complex Gaussian matrices; θ = 0.76 for quaternionic Gaussian matrices;

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Subexponential bounds for Barvinok’s estimator

Identity matrix: multiplicative error at least exp(cn) w.h.p. Matrix of all ones: multiplicative error at most exp(C√log n) (Goodman, 1963). What happens for other matrices?

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Subexponential bounds for Barvinok’s estimator

Identity matrix: multiplicative error at least exp(cn) w.h.p. Matrix of all ones: multiplicative error at most exp(C√log n) (Goodman, 1963). What happens for other matrices? Balanced entries (Friedland, Rider, Zeitouni, 2004): if max ai,j ≤ t · min ai,j, then e−o(n) ≤ det2(G ⊙ A1/2) perm(A) ≤ eo(n) with probability 1 − o(1) as n → ∞. The bound is asymptotic.

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Balanced entries

Theorem (Costello, Vu, 2009)

If max ai,j ≤ t · min ai,j, then exp

• −O(n2/3 log n)
• ≤ det2(G ⊙ A1/2)

perm(A) ≤ exp

• O(n2/3 log n)
• with probability 1 − n−C.

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Balanced entries

Theorem (Costello, Vu, 2009)

If max ai,j ≤ t · min ai,j, then exp

• −O(n2/3 log n)
• ≤ det2(G ⊙ A1/2)

perm(A) ≤ exp

• O(n2/3 log n)
• with probability 1 − n−C.

Not applicable for matrices with zeros. Linial–Samorodnitsky–Wigderson algorithm estimates the permanent with polynomial error for balanced matrices.

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Question: for which graphs would Barvinok’s estimator yield a small error?

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Broadly connected bipartite graphs

Let Γ = (L, R, V) be an n × n bipartite graph.

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Broadly connected bipartite graphs

Let Γ = (L, R, V) be an n × n bipartite graph. A vertex i ∈ L is δ-broadly connected to a set J ⊂ R if it is connected to at least (δ/2) · |J| vertices of J. J

Mark Rudelson (Michigan) Permanent estimators via random matrices 11 / 25

Broadly connected bipartite graphs

Let Γ = (L, R, V) be an n × n bipartite graph. A vertex i ∈ L is δ-broadly connected to a set J ⊂ R if it is connected to at least (δ/2) · |J| vertices of J.

Broadly connected graph

Let δ, κ > 0, δ/2 > κ. The graph Γ is (δ, κ)-broadly connected if

1

Left degree condition: deg(i) ≥ δn for all i ∈ [n];

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Broadly connected bipartite graphs

Let Γ = (L, R, V) be an n × n bipartite graph. A vertex i ∈ L is δ-broadly connected to a set J ⊂ R if it is connected to at least (δ/2) · |J| vertices of J.

Broadly connected graph

Let δ, κ > 0, δ/2 > κ. The graph Γ is (δ, κ)-broadly connected if

1

Left degree condition: deg(i) ≥ δn for all i ∈ [n];

2

Right degree condition: deg(j) ≥ δn for all j ∈ [n];

Mark Rudelson (Michigan) Permanent estimators via random matrices 11 / 25

Broadly connected bipartite graphs

Let Γ = (L, R, V) be an n × n bipartite graph. A vertex i ∈ L is δ-broadly connected to a set J ⊂ R if it is connected to at least (δ/2) · |J| vertices of J.

Broadly connected graph

Let δ, κ > 0, δ/2 > κ. The graph Γ is (δ, κ)-broadly connected if

1

Left degree condition: deg(i) ≥ δn for all i ∈ [n];

2

Right degree condition: deg(j) ≥ δn for all j ∈ [n];

3

Strong expansion condition: for any set J ⊂ [m] the set of its δ-broadly connected neighbors has the cardinality |I(J)| ≥ min

• (1 + κ)|J|, n
• .

J I(J)

Mark Rudelson (Michigan) Permanent estimators via random matrices 11 / 25

Broadly connected bipartite graphs

Let Γ = (L, R, V) be an n × n bipartite graph. A vertex i ∈ L is δ-broadly connected to a set J ⊂ R if it is connected to at least (δ/2) · |J| vertices of J.

Broadly connected graph

Let δ, κ > 0, δ/2 > κ. The graph Γ is (δ, κ)-broadly connected if

1

Left degree condition: deg(i) ≥ δn for all i ∈ [n];

2

Right degree condition: deg(j) ≥ δn for all j ∈ [n];

3

Strong expansion condition: for any set J ⊂ [m] the set of its δ-broadly connected neighbors has the cardinality |I(J)| ≥ min

• (1 + κ)|J|, n
• .

J I(J)

Mark Rudelson (Michigan) Permanent estimators via random matrices 11 / 25

Broadly connected bipartite graphs

Let Γ = (L, R, V) be an n × n bipartite graph. A vertex i ∈ L is δ-broadly connected to a set J ⊂ R if it is connected to at least (δ/2) · |J| vertices of J.

Broadly connected graph

Let δ, κ > 0, δ/2 > κ. The graph Γ is (δ, κ)-broadly connected if

1

Left degree condition: deg(i) ≥ δn for all i ∈ [n];

2

Right degree condition: deg(j) ≥ δn for all j ∈ [n];

3

Strong expansion condition: for any set J ⊂ [m] the set of its δ-broadly connected neighbors has the cardinality |I(J)| ≥ min

• (1 + κ)|J|, n
• .

J I(J)

Mark Rudelson (Michigan) Permanent estimators via random matrices 11 / 25

Broadly connected bipartite graphs

Let Γ = (L, R, V) be an n × n bipartite graph. A vertex i ∈ L is δ-broadly connected to a set J ⊂ R if it is connected to at least (δ/2) · |J| vertices of J.

Broadly connected graph

Let δ, κ > 0, δ/2 > κ. The graph Γ is (δ, κ)-broadly connected if

1

Left degree condition: deg(i) ≥ δn for all i ∈ [n];

2

Right degree condition: deg(j) ≥ δn for all j ∈ [n];

3

Strong expansion condition: for any set J ⊂ [m] the set of its δ-broadly connected neighbors has the cardinality |I(J)| ≥ min

• (1 + κ)|J|, n
• .

J I(J)

Mark Rudelson (Michigan) Permanent estimators via random matrices 11 / 25

Concentration for Barvinok’s estimator: how strong?

Good news: the Barvinok estimator is strongly concentrated: the multiplicative error is O(exp

• (cn log n)1/3

with high probability

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Concentration for Barvinok’s estimator: how strong?

Good news: the Barvinok estimator is strongly concentrated: the multiplicative error is O(exp

• (cn log n)1/3

with high probability Bad news: It can be concentrated around a wrong value

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Concentration for Barvinok’s estimator: how strong?

Good news: the Barvinok estimator is strongly concentrated: the multiplicative error is O(exp

• (cn log n)1/3

with high probability Bad news: It can be concentrated around a wrong value

Theorem (R’–Zeitouni, 2013)

Let A be the adjacency matrix A of an n × n broadly connected bipartite graph. Let Γ be a random Gaussian matrix with variance matrix A. Then for any τ ≥ 1 P

• ≤ det2(Γ)

M ≤

• ≥ 1 − small

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Concentration for Barvinok’s estimator: how strong?

Good news: the Barvinok estimator is strongly concentrated: the multiplicative error is O(exp

• (cn log n)1/3

with high probability Bad news: It can be concentrated around a wrong value

Theorem (R’–Zeitouni, 2013)

Let A be the adjacency matrix A of an n × n broadly connected bipartite graph. Let Γ be a random Gaussian matrix with variance matrix A. Then for any τ ≥ 1 P

• exp
• −C(τn log n)1/3

≤ det2(Γ) M ≤ exp

• C(τn log n)1/3

≥ 1 − small

Mark Rudelson (Michigan) Permanent estimators via random matrices 12 / 25

Concentration for Barvinok’s estimator: how strong?

Good news: the Barvinok estimator is strongly concentrated: the multiplicative error is O(exp

• (cn log n)1/3

with high probability Bad news: It can be concentrated around a wrong value

Theorem (R’–Zeitouni, 2013)

Let A be the adjacency matrix A of an n × n broadly connected bipartite graph. Let Γ be a random Gaussian matrix with variance matrix A. Then for any τ ≥ 1 P

• exp
• −C(τn log n)1/3

≤ det2(Γ) M ≤ exp

• C(τn log n)1/3

≥ 1 − exp(−τ)+ exp

• −c√n/ log n
• Mark Rudelson (Michigan)

Permanent estimators via random matrices 12 / 25

Concentration for Barvinok’s estimator: how strong?

Good news: the Barvinok estimator is strongly concentrated: the multiplicative error is O(exp

• (cn log n)1/3

with high probability Bad news: It can be concentrated around a wrong value

Theorem (R’–Zeitouni, 2013)

Let A be the adjacency matrix A of an n × n broadly connected bipartite graph. Let Γ be a random Gaussian matrix with variance matrix A. Then for any τ ≥ 1 P

• exp
• −C(τn log n)1/3

≤ det2(Γ) M ≤ exp

• C(τn log n)1/3

≥ 1 − exp(−τ)+ exp

• −c√n/ log n
• and

exp

• −C(n log n)1/2

≤ M perm(A) ≤ 1.

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Concentration for Barvinok’s estimator: where?

Theorem (R’–Zeitouni, 2013)

Let A be the adjacency matrix A of an n × n broadly connected bipartite graph. Let Γ be a random Gaussian matrix with variance matrix A. Then for any τ ≥ 1 P

• exp
• −C(τn log n)1/3

≤ det2(Γ) M ≤ exp

• C(τn log n)1/3

≥ 1 − exp(−τ)+ exp

• −c√n/ log n
• and

exp

• −C(n log n)1/2

≤ M perm(A) ≤ 1. perm(A) = E det2(Γ);

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Concentration for Barvinok’s estimator: where?

Theorem (R’–Zeitouni, 2013)

Let A be the adjacency matrix A of an n × n broadly connected bipartite graph. Let Γ be a random Gaussian matrix with variance matrix A. Then for any τ ≥ 1 P

• exp
• −C(τn log n)1/3

≤ det2(Γ) M ≤ exp

• C(τn log n)1/3

≥ 1 − exp(−τ)+ exp

• −c√n/ log n
• and

exp

• −C(n log n)1/2

≤ M perm(A) ≤ 1. perm(A) = E det2(Γ); M = exp

• E log det2(Γ)
• .

Mark Rudelson (Michigan) Permanent estimators via random matrices 13 / 25

Concentration for Barvinok’s estimator: where?

Theorem (R’–Zeitouni, 2013)

Let A be the adjacency matrix A of an n × n broadly connected bipartite graph. Let Γ be a random Gaussian matrix with variance matrix A. Then for any τ ≥ 1 P

• exp
• −C(τn log n)1/3

≤ det2(Γ) M ≤ exp

• C(τn log n)1/3

≥ 1 − exp(−τ)+ exp

• −c√n/ log n
• and

exp

• −C(n log n)1/2

≤ M perm(A) ≤ 1. perm(A) = E det2(Γ); M = exp

• E log det2(Γ)
• .

det is a highly non-linear function ⇒ det(Γ) has heavy tails.

Mark Rudelson (Michigan) Permanent estimators via random matrices 13 / 25

Results for matrices

Large entries graph

Let s > 0 and let B be an n × n matrix B with non-negative entries. Define the bipartite graph ΓB(s) connecting the vertices i and j whenever bi,j ≥ s

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Results for matrices

Large entries graph

Let s > 0 and let B be an n × n matrix B with non-negative entries. Define the bipartite graph ΓB(s) connecting the vertices i and j whenever bi,j ≥ s B =     0.7 0.1 0.5 0.1 0.6 0.8 0.2 0.6 0.6 0.3 0.5 0.2 0.8 0.7 0.3     ⇒     1 1 1 1 1 1 1 1 1     (s = 0.5) Consider matrices with broadly connected large entries graphs.

Mark Rudelson (Michigan) Permanent estimators via random matrices 14 / 25

Results for matrices

Theorem

Let B be an n × n matrix such that

n

• i=1

bi,j ≤ 1 for all j ∈ [n]; and

n

• j=1

bi,j ≤ 1 for all i ∈ [n], and 0 ≤ bi,j ≤ bn/n, where 0 < bn ≤ n. Assume that the large entries graph ΓB(1/n) is broadly connected. Let Γ be a random Gaussian matrix with variance matrix A.

Mark Rudelson (Michigan) Permanent estimators via random matrices 15 / 25

Results for matrices

Theorem

Let B be an n × n matrix such that

n

• i=1

bi,j ≤ 1 for all j ∈ [n]; and

n

• j=1

bi,j ≤ 1 for all i ∈ [n], and 0 ≤ bi,j ≤ bn/n, where 0 < bn ≤ n. Assume that the large entries graph ΓB(1/n) is broadly connected. Let Γ be a random Gaussian matrix with variance matrix A. Then for any τ ≥ 1 P

• exp
• −C(τbnn)1/3logc n
• ≤ det2(Γ)

M ≤ exp

• C(τbnn)1/3logc n
• ≥ 1 − exp(−τ)+ exp
• −c√n/ logc n
• Mark Rudelson (Michigan)

Permanent estimators via random matrices 15 / 25

Results for matrices

Theorem

Let B be an n × n matrix such that

n

• i=1

bi,j ≤ 1 for all j ∈ [n]; and

n

• j=1

bi,j ≤ 1 for all i ∈ [n], and 0 ≤ bi,j ≤ bn/n, where 0 < bn ≤ n. Assume that the large entries graph ΓB(1/n) is broadly connected. Let Γ be a random Gaussian matrix with variance matrix A. Then for any τ ≥ 1 P

• exp
• −C(τbnn)1/3logc n
• ≤ det2(Γ)

M ≤ exp

• C(τbnn)1/3logc n
• ≥ 1 − exp(−τ)+ exp
• −c√n/ logc n
• and

exp

• −C(bnn)1/2logc n
• ≤

M perm(A) ≤ 1.

Mark Rudelson (Michigan) Permanent estimators via random matrices 15 / 25

Results for matrices

Theorem

P

• exp
• −C(τbnn)1/3logc n
• ≤ det2(Γ)

M ≤ exp

• C(τbnn)1/3logc n
• ≥ 1 − exp(−τ)+ exp
• −c√n/ logc n
• and

exp

• −C(bnn)1/2logc n
• ≤

M perm(A) ≤ 1. Small maximal entry: max bi,j = o(1) or bn = o(n):

Barvinok’s estimator is well-concentrated about the permanent.

Mark Rudelson (Michigan) Permanent estimators via random matrices 16 / 25

Results for matrices

Theorem

P

• exp
• −C(τbnn)1/3logc n
• ≤ det2(Γ)

M ≤ exp

• C(τbnn)1/3logc n
• ≥ 1 − exp(−τ)+ exp
• −c√n/ logc n
• and

exp

• −C(bnn)1/2logc n
• ≤

M perm(A) ≤ 1. Small maximal entry: max bi,j = o(1) or bn = o(n):

Barvinok’s estimator is well-concentrated about the permanent.

Large maximal entry: max bi,j = Ω(1) or bn = n · max bi,j = Ω(n):

Barvinok’s estimator is well-concentrated: (τbnn)1/3 = O(n2/3); It may be concentrated exponentially far from the permanent: √bnn = Ω(n). Consistent failure is possible.

Mark Rudelson (Michigan) Permanent estimators via random matrices 16 / 25

Example of a consistent failure

Let B be the n × n matrix with entries bi,j =

• θ

if i = j

1−θ n−1

if i = j . The matrix B is doubly stochastic for θ ∈ (0, 1). B has no zero entries. ΓB is a complete bipartite graph.

Mark Rudelson (Michigan) Permanent estimators via random matrices 17 / 25

Example of a consistent failure

Let B be the n × n matrix with entries bi,j =

• θ

if i = j

1−θ n−1

if i = j . The matrix B is doubly stochastic for θ ∈ (0, 1). B has no zero entries. ΓB is a complete bipartite graph.

Theorem

There exists θ0 < 1 such that for any θ ∈ (θ0, 1) det2(B1/2 ⊙ G) < e−cn perm(B) with high probability.

Mark Rudelson (Michigan) Permanent estimators via random matrices 17 / 25

Approach to concentration

Aim: X(G) := det2(A1/2 ⊙ G) is concentrated. det2(A1/2 ⊙ G) is highly non-linear ⇒ log(det2(A1/2 ⊙ G)) is easier to control. Modified aim : Y(G) = log det2(A1/2 ⊙ G) is concentrated around its expectation. We will have to compare the concentration for X(G) and Y(G) at the end.

Mark Rudelson (Michigan) Permanent estimators via random matrices 18 / 25

Approach to concentration

Aim: X(G) := det2(A1/2 ⊙ G) is concentrated. det2(A1/2 ⊙ G) is highly non-linear ⇒ log(det2(A1/2 ⊙ G)) is easier to control. Modified aim : Y(G) = log det2(A1/2 ⊙ G) is concentrated around its expectation. We will have to compare the concentration for X(G) and Y(G) at the end. There exists a subgaussian concentration inequality for Lipschitz functions on Rn×n with respect to the gaussian measure: P (|F(G) − EF(G)| ≥ t) ≤ 2 exp

• −

t2 2L2(F)

• .

Mark Rudelson (Michigan) Permanent estimators via random matrices 18 / 25

Approach to concentration

Aim: X(G) := det2(A1/2 ⊙ G) is concentrated. det2(A1/2 ⊙ G) is highly non-linear ⇒ log(det2(A1/2 ⊙ G)) is easier to control. Modified aim : Y(G) = log det2(A1/2 ⊙ G) is concentrated around its expectation. We will have to compare the concentration for X(G) and Y(G) at the end. There exists a subgaussian concentration inequality for Lipschitz functions on Rn×n with respect to the gaussian measure: P (|F(G) − EF(G)| ≥ t) ≤ 2 exp

• −

t2 2L2(F)

• .

log det2(A1/2 ⊙ G) is not Lipschitz.

Mark Rudelson (Michigan) Permanent estimators via random matrices 18 / 25

What is non-Lipschitz in log determinant?

log det2(A1/2 ⊙ G) = 2 n

j=1 log sj(A1/2 ⊙ G).

The maping G → A1/2 ⊙ G is Lipschitz. The mapping M →

• s1(M), . . . , sn(M)
• is Lipschitz.

Logarithm is not a Lipschitz function.

Mark Rudelson (Michigan) Permanent estimators via random matrices 19 / 25

What is non-Lipschitz in log determinant?

log det2(A1/2 ⊙ G) = 2 n

j=1 log sj(A1/2 ⊙ G).

The maping G → A1/2 ⊙ G is Lipschitz. The mapping M →

• s1(M), . . . , sn(M)
• is Lipschitz.

Truncated logarithm logε x = max(log x, log ε) is a Lipschitz function. We have to guarantee that sn(A1/2 ⊙ G) > ε with high probability.

Mark Rudelson (Michigan) Permanent estimators via random matrices 19 / 25

Estimating the log determinant

1

Use non-asymptotic random matrix theory to show that P (sn(A1/2 ⊙ G) > ε) = 1 − o(1);

2

Replace log det2(A1/2 ⊙ G) = 2

n

• j=1

log sj(A1/2 ⊙ G) by logε det2(A1/2 ⊙ G) = 2

n

• j=1

logε sj(A1/2 ⊙ G) (holds with high probability);

3

Establish concentration for logε det2(A1/2 ⊙ G).

Mark Rudelson (Michigan) Permanent estimators via random matrices 20 / 25

Estimating the log determinant: first attempt

1

Use non-asymptotic random matrix theory to show that P (sn(A1/2 ⊙ G) > ε) = 1 − o(1);

2

Replace log det2(A1/2 ⊙ G) = 2

n

• j=1

log sj(A1/2 ⊙ G) by logε det2(A1/2 ⊙ G) = 2

n

• j=1

logε sj(A1/2 ⊙ G) (holds with high probability);

3

Establish concentration for logε det2(A1/2 ⊙ G). ← Fails

Mark Rudelson (Michigan) Permanent estimators via random matrices 20 / 25

Estimating the log determinant: first attempt

1

Use non-asymptotic random matrix theory to show that P (sn(A1/2 ⊙ G) > ε) = 1 − o(1);

2

Replace log det2(A1/2 ⊙ G) = 2

n

• j=1

log sj(A1/2 ⊙ G) by logε det2(A1/2 ⊙ G) = 2

n

• j=1

logε sj(A1/2 ⊙ G) (holds with high probability);

3

Establish concentration for logε det2(A1/2 ⊙ G). ← Fails

Reasons for the failure

The least singular value is too small ⇒ the Lipschitz constant is too big. The least singular value is not concentrated at all.

Mark Rudelson (Michigan) Permanent estimators via random matrices 20 / 25

Estimating the log determinant

1

Non-asymptotic random matrix theory ⇒ Bound for intermediate singular values: sj(A1/2 ⊙ G) ≥ εj with high probability;

Mark Rudelson (Michigan) Permanent estimators via random matrices 21 / 25

Estimating the log determinant

1

Non-asymptotic random matrix theory ⇒ Bound for intermediate singular values: sj(A1/2 ⊙ G) ≥ εj with high probability;

2

Intermediate singular values:

Mark Rudelson (Michigan) Permanent estimators via random matrices 21 / 25

Estimating the log determinant

1

Non-asymptotic random matrix theory ⇒ Bound for intermediate singular values: sj(A1/2 ⊙ G) ≥ εj with high probability;

2

Intermediate singular values:

Larger than the last one ⇒ stronger concentration for truncated logarithm;

Mark Rudelson (Michigan) Permanent estimators via random matrices 21 / 25

Estimating the log determinant

1

Non-asymptotic random matrix theory ⇒ Bound for intermediate singular values: sj(A1/2 ⊙ G) ≥ εj with high probability;

2

Intermediate singular values:

Larger than the last one ⇒ stronger concentration for truncated logarithm; Stronger concentrated.

Mark Rudelson (Michigan) Permanent estimators via random matrices 21 / 25

Estimating the log determinant

1

Non-asymptotic random matrix theory ⇒ Bound for intermediate singular values: sj(A1/2 ⊙ G) ≥ εj with high probability;

2

Intermediate singular values:

Larger than the last one ⇒ stronger concentration for truncated logarithm; Stronger concentrated.

3

log det2(A1/2 ⊙ G) = n−k

j=1 log sj(A1/2 ⊙ G) + n j=n−k+1 log sj(A1/2 ⊙ G)

Mark Rudelson (Michigan) Permanent estimators via random matrices 21 / 25

Estimating the log determinant

1

Non-asymptotic random matrix theory ⇒ Bound for intermediate singular values: sj(A1/2 ⊙ G) ≥ εj with high probability;

2

Intermediate singular values:

Larger than the last one ⇒ stronger concentration for truncated logarithm; Stronger concentrated.

3

log det2(A1/2 ⊙ G) = n−k

j=1 log sj(A1/2 ⊙ G) + n j=n−k+1 log sj(A1/2 ⊙ G)

4

n−k

j=1 log sj(A1/2 ⊙ G) = n−k j=1 logε sj(A1/2 ⊙ G) is a (1/ε)-Lipschitz function

⇒ we can use concentration for the first singular values

Mark Rudelson (Michigan) Permanent estimators via random matrices 21 / 25

Estimating the log determinant

1

Non-asymptotic random matrix theory ⇒ Bound for intermediate singular values: sj(A1/2 ⊙ G) ≥ εj with high probability;

2

Intermediate singular values:

Larger than the last one ⇒ stronger concentration for truncated logarithm; Stronger concentrated.

3

log det2(A1/2 ⊙ G) = n−k

j=1 log sj(A1/2 ⊙ G) + n j=n−k+1 log sj(A1/2 ⊙ G)

4

n−k

j=1 log sj(A1/2 ⊙ G) = n−k j=1 logε sj(A1/2 ⊙ G) is a (1/ε)-Lipschitz function

⇒ we can use concentration for the first singular values

5

• n

j=n−k+1 log sj(A1/2 ⊙ G)

• ≤ (n − k)| log αn|

⇒ contribution of the last singular values is limited.

Mark Rudelson (Michigan) Permanent estimators via random matrices 21 / 25

Estimating the log determinant: second attempt

1

Non-asymptotic random matrix theory ⇒ Bound for intermediate singular values: sj(A1/2 ⊙ G) ≥ εj with high probability;

2

Intermediate singular values:

Larger than the last one ⇒ stronger concentration for truncated logarithm; Stronger concentrated.

3

log det2(A1/2 ⊙ G) = n−k

j=1 log sj(A1/2 ⊙ G) + n j=n−k+1 log sj(A1/2 ⊙ G)

4

n−k

j=1 log sj(A1/2 ⊙ G) = n−k j=1 logε sj(A1/2 ⊙ G) is a (1/ε)-Lipschitz function

⇒ we can use concentration for the first singular values

5

• n

j=n−k+1 log sj(A1/2 ⊙ G)

• ≤ (n − k)| log αn|

⇒ contribution of the last singular values is limited. A single threshold is not enough to get a fine picture.

Mark Rudelson (Michigan) Permanent estimators via random matrices 21 / 25

Estimating the log determinant

1

Non-asymptotic random matrix theory ⇒ Bound for intermediate singular values: sj(A1/2 ⊙ G) ≥ εj with high probability;

2

log det2(A1/2 ⊙ G) = n−k

j=1 log sj(A1/2 ⊙ G) + n j=n−k+1 log sj(A1/2 ⊙ G)

3

n−k

j=1 log sj(A1/2 ⊙ G) = n−k j=1 logε sj(A1/2 ⊙ G) is a (1/ε)-Lipschitz function

⇒ we can use concentration for the first singular values

4

• n

j=n−k+1 log sj(A1/2 ⊙ G)

• ≤ (n − k)| log αn|

⇒ contribution of the last singular values is limited.

Mark Rudelson (Michigan) Permanent estimators via random matrices 22 / 25

Estimating the log determinant

1

Non-asymptotic random matrix theory ⇒ Bound for intermediate singular values: sj(A1/2 ⊙ G) ≥ εj with high probability;

2

log det2(A1/2 ⊙ G) = n−k

j=1 log sj(A1/2 ⊙ G) + n j=n−k+1 log sj(A1/2 ⊙ G)

3

n−k

j=1 log sj(A1/2 ⊙ G) = n−k j=1 logεj sj(A1/2 ⊙ G) is a (?)-Lipschitz function ⇒

balance the concentration

4

• n

j=n−k+1 log sj(A1/2 ⊙ G)

• ≤ (n − k)| log αn|

⇒ contribution of the last singular values is limited.

Mark Rudelson (Michigan) Permanent estimators via random matrices 22 / 25

Estimating the log determinant

1

Non-asymptotic random matrix theory ⇒ Bound for intermediate singular values: sj(A1/2 ⊙ G) ≥ εj with high probability;

2

log det2(A1/2 ⊙ G) = n−k

j=1 log sj(A1/2 ⊙ G) + n j=n−k+1 log sj(A1/2 ⊙ G)

3

n−k

j=1 log sj(A1/2 ⊙ G) = n−k j=1 logεj sj(A1/2 ⊙ G) is a (?)-Lipschitz function ⇒

balance the concentration

4

• n

j=n−k+1 log sj(A1/2 ⊙ G)

• ≤ n

j=n−k+1 | log εj|

⇒ contribution of the last singular values is limited.

Mark Rudelson (Michigan) Permanent estimators via random matrices 22 / 25

Estimating the log determinant

1

Non-asymptotic random matrix theory ⇒ Bound for intermediate singular values: sj(A1/2 ⊙ G) ≥ εj with high probability;

2

log det2(A1/2 ⊙ G) = n−k

j=1 log sj(A1/2 ⊙ G) + n j=n−k+1 log sj(A1/2 ⊙ G)

3

n−k

j=1 log sj(A1/2 ⊙ G) = n−k j=1 logεj sj(A1/2 ⊙ G) is a (?)-Lipschitz function ⇒

balance the concentration

4

• n

j=n−k+1 log sj(A1/2 ⊙ G)

• ≤ n

j=n−k+1 | log εj|

⇒ contribution of the last singular values is limited.

5

How to choose the threshold k? Smaller k ⇒ smaller error. Larger k ⇒ stronger concentration.

Mark Rudelson (Michigan) Permanent estimators via random matrices 22 / 25

Choosing the right threshold

log det2(A1/2 ⊙ G) =

n−k

• j=1

logεj sj(A1/2 ⊙ G) +

n

• j=n−k+1

error terms

Mark Rudelson (Michigan) Permanent estimators via random matrices 23 / 25

Choosing the right threshold

log det2(A1/2 ⊙ G) = log det2(A1/2 ⊙ G) +

n

• j=n−k+1

error terms

• log det2(A1/2 ⊙ G) is concentrated about its expectation.

Smaller k ⇒ smaller error. Larger k ⇒ stronger concentration.

Mark Rudelson (Michigan) Permanent estimators via random matrices 23 / 25

Choosing the right threshold

log det2(A1/2 ⊙ G) = log det2(A1/2 ⊙ G) +

n

• j=n−k+1

error terms

• log det2(A1/2 ⊙ G) is concentrated about its expectation.

Smaller k ⇒ smaller error. log det2(A1/2 ⊙ G) is close to E log det2(A1/2 ⊙ G) with high probability. This may be far from log perm(A). Larger k ⇒ stronger concentration.

Mark Rudelson (Michigan) Permanent estimators via random matrices 23 / 25

Choosing the right threshold

log det2(A1/2 ⊙ G) = log det2(A1/2 ⊙ G) +

n

• j=n−k+1

error terms

• log det2(A1/2 ⊙ G) is concentrated about its expectation.

Smaller k ⇒ smaller error. log det2(A1/2 ⊙ G) is close to E log det2(A1/2 ⊙ G) with high probability. This may be far from log perm(A). Larger k ⇒ stronger concentration. Strong concentration ⇒ E log det2(A1/2 ⊙ G) ≈ log E det2(A1/2 ⊙ G)

Mark Rudelson (Michigan) Permanent estimators via random matrices 23 / 25

Choosing the right threshold

log det2(A1/2 ⊙ G) = log det2(A1/2 ⊙ G) +

n

• j=n−k+1

error terms

• log det2(A1/2 ⊙ G) is concentrated about its expectation.

Smaller k ⇒ smaller error. log det2(A1/2 ⊙ G) is close to E log det2(A1/2 ⊙ G) with high probability. This may be far from log perm(A). Larger k ⇒ stronger concentration. Strong concentration ⇒ E log det2(A1/2 ⊙ G) ≈ log E det2(A1/2 ⊙ G) = log perm(A) up to the error terms.

Mark Rudelson (Michigan) Permanent estimators via random matrices 23 / 25

Conclusion

Barvinok’s estimator is always strongly concentrated for broadly connected graphs and matrices.

Mark Rudelson (Michigan) Permanent estimators via random matrices 24 / 25

Conclusion

Barvinok’s estimator is always strongly concentrated for broadly connected graphs and matrices. It is concentrated around exp E log det2(A1/2 ⊙ G) ⇒ the main sourse of errors is deterministic.

Mark Rudelson (Michigan) Permanent estimators via random matrices 24 / 25

Conclusion

Barvinok’s estimator is always strongly concentrated for broadly connected graphs and matrices. It is concentrated around exp E log det2(A1/2 ⊙ G) ⇒ the main sourse of errors is deterministic. The gap between exp E log det2(A1/2 ⊙ G) and the permanent is subexponential for graphs

Mark Rudelson (Michigan) Permanent estimators via random matrices 24 / 25

Conclusion

Barvinok’s estimator is always strongly concentrated for broadly connected graphs and matrices. It is concentrated around exp E log det2(A1/2 ⊙ G) ⇒ the main sourse of errors is deterministic. The gap between exp E log det2(A1/2 ⊙ G) and the permanent is subexponential for graphs and for almost doubly stochastic matrices in the subcritical case max ai,j = o(1).

Mark Rudelson (Michigan) Permanent estimators via random matrices 24 / 25

Conclusion

Barvinok’s estimator is always strongly concentrated for broadly connected graphs and matrices. It is concentrated around exp E log det2(A1/2 ⊙ G) ⇒ the main sourse of errors is deterministic. The gap between exp E log det2(A1/2 ⊙ G) and the permanent is subexponential for graphs and for almost doubly stochastic matrices in the subcritical case max ai,j = o(1). It may be exponential in the critical case max ai,j = Ω(1) ⇒ consistent failure.

Mark Rudelson (Michigan) Permanent estimators via random matrices 24 / 25