Quantitative invertibility of random matrices: a combinatorial perspective Vishesh Jain Massachusetts Institute of Technology Stanford Combinatorics Seminar October 31, 2019 Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 1 / 36
The quantitative invertibility problem Definition (Least singular value) The least singular value of an n × n matrix M n is defined by s n ( M n ) := v ∈ S n − 1 � M n v � 2 . inf Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 2 / 36
The quantitative invertibility problem Definition (Least singular value) The least singular value of an n × n matrix M n is defined by s n ( M n ) := v ∈ S n − 1 � M n v � 2 . inf Quantitative invertibility problem What is the probability that s n ( M n ) is smaller than η ≥ 0? Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 2 / 36
Regime I: Invertibility of random discrete matrices Suppose that each entry of M n is an independent Rademacher random variable i.e. +1 or − 1 with probability 1 / 2 each. Estimate Pr( s n ( M n ) = 0). Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 3 / 36
Regime I: Invertibility of random discrete matrices Suppose that each entry of M n is an independent Rademacher random variable i.e. +1 or − 1 with probability 1 / 2 each. Estimate Pr( s n ( M n ) = 0). Folklore Conjecture: Pr( s n ( M n ) = 0) � n 2 2 − n . Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 3 / 36
Regime I: Invertibility of random discrete matrices Suppose that each entry of M n is an independent Rademacher random variable i.e. +1 or − 1 with probability 1 / 2 each. Estimate Pr( s n ( M n ) = 0). Folklore Conjecture: Pr( s n ( M n ) = 0) � n 2 2 − n . Koml´ os (1967): Pr ( s n ( M n ) = 0) = o n (1). Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 3 / 36
Regime I: Invertibility of random discrete matrices Suppose that each entry of M n is an independent Rademacher random variable i.e. +1 or − 1 with probability 1 / 2 each. Estimate Pr( s n ( M n ) = 0). Folklore Conjecture: Pr( s n ( M n ) = 0) � n 2 2 − n . Koml´ os (1967): Pr ( s n ( M n ) = 0) = o n (1). edi (1995): Pr( s n ( M n ) = 0) � 0 . 999 n . Kahn, Koml´ os, and Szemer´ Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 3 / 36
Regime I: Invertibility of random discrete matrices Suppose that each entry of M n is an independent Rademacher random variable i.e. +1 or − 1 with probability 1 / 2 each. Estimate Pr( s n ( M n ) = 0). Folklore Conjecture: Pr( s n ( M n ) = 0) � n 2 2 − n . Koml´ os (1967): Pr ( s n ( M n ) = 0) = o n (1). edi (1995): Pr( s n ( M n ) = 0) � 0 . 999 n . Kahn, Koml´ os, and Szemer´ Tao and Vu (2006, 2007): Pr( s n ( M n ) = 0) � 0 . 75 n . Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 3 / 36
Regime I: Invertibility of random discrete matrices Suppose that each entry of M n is an independent Rademacher random variable i.e. +1 or − 1 with probability 1 / 2 each. Estimate Pr( s n ( M n ) = 0). Folklore Conjecture: Pr( s n ( M n ) = 0) � n 2 2 − n . Koml´ os (1967): Pr ( s n ( M n ) = 0) = o n (1). edi (1995): Pr( s n ( M n ) = 0) � 0 . 999 n . Kahn, Koml´ os, and Szemer´ Tao and Vu (2006, 2007): Pr( s n ( M n ) = 0) � 0 . 75 n . √ 2) n . Bourgain, Vu, and Wood (2010): Pr( s n ( M n ) = 0) � (1 / Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 3 / 36
Regime I: Invertibility of random discrete matrices Suppose that each entry of M n is an independent Rademacher random variable i.e. +1 or − 1 with probability 1 / 2 each. Estimate Pr( s n ( M n ) = 0). Folklore Conjecture: Pr( s n ( M n ) = 0) � n 2 2 − n . Koml´ os (1967): Pr ( s n ( M n ) = 0) = o n (1). edi (1995): Pr( s n ( M n ) = 0) � 0 . 999 n . Kahn, Koml´ os, and Szemer´ Tao and Vu (2006, 2007): Pr( s n ( M n ) = 0) � 0 . 75 n . √ 2) n . Bourgain, Vu, and Wood (2010): Pr( s n ( M n ) = 0) � (1 / Tikhomirov (2018): Pr( s n ( M n ) = 0) � (0 . 5 + o n (1)) n . Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 3 / 36
Regime II: Least singular value of ‘typical’ random matrices Suppose that each entry of M n is an independent copy of the standard Gaussian. Estimate s n ( M n ) for a ‘typical’ such matrix. Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 4 / 36
Regime II: Least singular value of ‘typical’ random matrices Suppose that each entry of M n is an independent copy of the standard Gaussian. Estimate s n ( M n ) for a ‘typical’ such matrix. Edelman (1988), Szarek (1991): Pr( s n ( M n ) ≤ ǫ n − 1 / 2 ) ≤ ǫ . Hence, for ‘most’ such matrices, s n ( M n ) = Ω( n − 1 / 2 ). Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 4 / 36
Regime II: Least singular value of ‘typical’ random matrices Suppose that each entry of M n is an independent copy of the standard Gaussian. Estimate s n ( M n ) for a ‘typical’ such matrix. Edelman (1988), Szarek (1991): Pr( s n ( M n ) ≤ ǫ n − 1 / 2 ) ≤ ǫ . Hence, for ‘most’ such matrices, s n ( M n ) = Ω( n − 1 / 2 ). Sankar, Spielman, and Teng (2006): Pr( s n ( A n + M n ) ≤ ǫ n − 1 / 2 ) � ǫ . Here, A n is an arbitrary square matrix. Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 4 / 36
The Spielman-Teng conjecture Conjecture (Spielman and Teng, ICM 2002) Suppose that the entries of M n are independent Rademacher random variables. There exists some constant c ∈ (0 , 1) such that for all η ≥ 0 , Pr ( s n ( M n ) ≤ η ) ≤ √ n η + c n . This combines ‘Gaussian behavior’ with the added possibility of singularity. Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 5 / 36
Resolution of the Spielman-Teng conjecture Theorem (Rudelson and Vershynin, 2007) Suppose that the entries of M n are i.i.d. subgaussian random variables with mean 0 and variance 1 . Then, there exists c ∈ (0 , 1) such that for all η ≥ 0 , Pr( s n ( M n ) ≤ η ) � √ n η + c n . Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 6 / 36
Resolution of the Spielman-Teng conjecture Theorem (Rudelson and Vershynin, 2007) Suppose that the entries of M n are i.i.d. subgaussian random variables with mean 0 and variance 1 . Then, there exists c ∈ (0 , 1) such that for all η ≥ 0 , Pr( s n ( M n ) ≤ η ) � √ n η + c n . Rebrova and Tikhomirov (2015): Removed subgaussian assumption. Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 6 / 36
Resolution of the Spielman-Teng conjecture Theorem (Rudelson and Vershynin, 2007) Suppose that the entries of M n are i.i.d. subgaussian random variables with mean 0 and variance 1 . Then, there exists c ∈ (0 , 1) such that for all η ≥ 0 , Pr( s n ( M n ) ≤ η ) � √ n η + c n . Rebrova and Tikhomirov (2015): Removed subgaussian assumption. Livshyts, Tikhomirov, and Vershynin (2019): Removed identically distributed assumption. Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 6 / 36
Least singular value of shifted i.i.d. matrices Thus far, the high-dimensional geometric methods used in the proofs of the previous results have failed to address the following important model of random matrices: Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 7 / 36
Least singular value of shifted i.i.d. matrices Thus far, the high-dimensional geometric methods used in the proofs of the previous results have failed to address the following important model of random matrices: M n := A n + N n , where A n is a ‘large’ fixed complex matrix, and N n is a random matrix, each of whose entries is an independent copy of a complex random variable of mean 0 and variance 1. Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 7 / 36
Least singular value of shifted i.i.d. matrices Thus far, the high-dimensional geometric methods used in the proofs of the previous results have failed to address the following important model of random matrices: M n := A n + N n , where A n is a ‘large’ fixed complex matrix, and N n is a random matrix, each of whose entries is an independent copy of a complex random variable of mean 0 and variance 1. Why is this important? Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 7 / 36
Least singular value of shifted i.i.d. matrices For the strong circular law , known reductions (Girko, 1984; Bai, 1997; Tao and Vu, 2008) show that we need to study s n ( M n ) for M n = z · Id n + N n √ n , with z ∈ C fixed. Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 8 / 36
Least singular value of shifted i.i.d. matrices For the strong circular law , known reductions (Girko, 1984; Bai, 1997; Tao and Vu, 2008) show that we need to study s n ( M n ) for M n = z · Id n + N n √ n , with z ∈ C fixed. For numerical linear algebra, the smoothed analysis program of Spielman and Teng (2001) considers M n = A n + N n , where N n represents the random ‘noise’ in the system. Vishesh Jain (MIT) Quantitative invertibility of random matrices October 31, 2019 8 / 36
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