Two Delightful Major Simplifications

simplify

Arguably mathematics is getting harder, although some people claim that also in the old times parts of it were hard and known only to a few experts before major simplifications had changed  matters. Let me report here about two recent remarkable simplifications of major theorems. I am thankful to Nati Linial who told me about the first and to Itai Benjamini and Gady Kozma who told me about the second. Enjoy!

Random regular graphs are nearly Ramanujan: Charles Bordenave gives a new proof of Friedman’s second eigenvalue Theorem and its extension to random lifts

Here is the paper. Abstract: It was conjectured by Alon and proved by Friedman that a random d-regular graph has nearly the largest possible spectral gap, more precisely, the largest absolute value of the non-trivial eigenvalues of its adjacency matrix is at most 2\sqrt{d-1} +o(1) with probability tending to one as the size of the graph tends to infinity. We give a new proof of this statement. We also study related questions on random n-lifts of graphs and improve a recent result by Friedman and Kohler.

A simple proof for the theorem of Aizenman and Barsky and of Menshikov. Hugo Duminil-Copin and Vincent Tassion give  a new proof of the sharpness of the phase transition for Bernoulli percolation on \mathbb Z^d

Here is the paper Abstract: We provide a new proof of the sharpness of the phase transition for nearest-neighbour Bernoulli percolation. More precisely, we show that – for p<p_c, the probability that the origin is connected by an open path to distance $n$ decays exponentially fast in $n$. – for p>p_c, the probability that the origin belongs to an infinite cluster satisfies the mean-field lower bound \theta(p)\ge\tfrac{p-p_c}{p(1-p_c)}. This note presents the argument of this paper by the same authors, which is valid for long-range Bernoulli percolation (and for the Ising model) on arbitrary transitive graphs in the simpler framework of nearest-neighbour Bernoulli percolation on \mathbb Z^d.

Influence, Threshold, and Noise

 

poster4.1

My dear friend Itai Benjamini told me that he won’t be able to make it to my Tuesday talk on influence, threshold, and noise, and asked if I already have  the slides. So it occurred to me that perhaps I can practice the lecture on you, my readers, not just with the slides (here they are) but also roughly what I plan to say, some additional info, and some pedagogical hesitations. Of course, remarks can be very helpful.

I can also briefly report that there are plenty of exciting things happening around that I would love to report about – hopefully later in my travel-free summer. One more thing: while chatting with Yuval Rabani and Daniel Spielman I realized that there are various exciting things happening in algorithms (and not reported so much in blogs). Much progress has been made on basic questions: TSP, Bin Packing, flows & bipartite matching, market equilibria, and k-servers, to mention a few, and also new directions and methods. I am happy to announce that Yuval kindly agreed to write here an algorithmic column from time to time, and Daniel is considering contributing a guest post as well.

The second AMS-IMU meeting

Since the early 70s, I have been a devoted participants in our annual meetings of the Israeli Mathematical Union (IMU), and this year we will have the second joint meeting with the American Mathematical Society (AMS). Here is the program. There are many exciting lectures. Let me mention that Eran Nevo, this year Erdős’ prize winner, will give a lecture about the g-conjecture. Congratulations, Eran! Among the 22 exciting special sessions there are a few related to combinatorics, and even one organized by me on Wednsday and Thursday.

Combinatorics
Contact person: Gil Kalai, gil.kalai@gmail.com
TAU, Dan David building, Room 103
 Wed, 10:50-11:30 Van H. Vu (Yale University) Real roots of random polynomials (abstract)
Wed, 11:40-12:20 Oriol Serra (Universitat Politecnica de Catalunya, Barcelona)  Arithmetic Removal Lemmas (abstract)
 Wed, 12:30-13:10 Tali Kaufman (Bar-Ilan University)  Bounded degree high dimensional expanders (abstract)
 Wed, 16:00-16:40 Rom Pinchasi (Technion)  On the union of arithmetic progressions (abstract)
Wed, 16:50-17:30  Isabella Novik (University of Washington, Seattle) Face numbers of balanced spheres, manifolds, and pseudomanifolds (abstract)
 Wed, 17:40-18:20 Edward Scheinerman (Johns Hopkins University, Baltimore) On Vertex, Edge, and Vertex-Edge Random Graphs (abstract)
 Thu, 9:20-10:00 Yael Tauman Kalai (MSR, New England) The Evolution of Proofs in Computer Science (abstract)
 Thu, 10:10-10:50  Irit Dinur (Weitzman Institute)  Lifting locally consistent solutions to global solutions (abstract)
 Thu, 11:00-11:40 Benny Sudakov (ETH, Zurich) The minimum number of nonnegative edges in hypergraphs (abstract)

 

And now for my own lecture.

Influence, Threshold, and Noise:

Continue reading

Analysis of Boolean Functions week 5 and 6

Lecture 7

First passage percolation

1)  Models of percolation.

We talked about percolation introduced by Broadbent and Hammersley in 1957. The basic model is a model of random subgraphs of a grid in n-dimensional space. (Other graphs were considered later as well.) Here, a grid is a graph whose vertices have integers coordinates and where two vertices are adjacent if their Euclidean distance is one. Every edge of the grid-graph is taken (or is “open” in the percolation jargon) with the same probability p, independently. We mentioned some basic questions – is there an infinite component? How many infinite components are there? What is the probability that the origin belongs to such an infinite component as a function of p?

I mentioned two results: The first  is Kesten’s celebrated result that the critical probability for planar percolation is 1/2. The other by Burton and Keane is that in very general situations almost surely there is a unique infinite component or none at all. This was a good point to mention a famous conjecture- The dying percolation conjecture (especially in dimension 3) which asserts that at the critical probability there is no infinite component.

We will come back to this basic model of percolation later in the course, but for now we moved to a related more recent model.

2) First passage percolation

We talked about first passage percolation introduced by Hammersley and Welsh in 1965. Again we consider the infinite graph of a grid and this time we let the length of every edge be 1 with probability 1/2 and 2 with probability 1/2 (independently). These weights describe a random metric on this infinite graph that we wish to understand. We consider two vertices (0,0) and (v,0) (for high dimension the second entry can account for a (d-1) dimensional vectors, but we can restrict our attention to d=2) and we let D(x) be the distance between these two vectors. We explained how D is an integer values function on a discrete cube with Liphshitz constant 1. The question we want to address is : What is the variance of D?

Why do we study the variance, when we do not know exactly the expectation, you may ask? (I remember Lerry Shepp asking this when I talked about it at Bell Labs in the early 90s.) One answer is that we know that the expectation of D is linear, and for the variance we do not know how it behaves. Second, we expect that telling the expectation precisely will depend on the model while the way the variance grows and perhaps D‘s limiting distribution, will be universal (say, for dimension 2). And third, we do not give up on the expectation as well. 

Here is what we showed:

1) From the inequality var(D)=\sum_{S\ne \emptyset}\hat D^2(S)\le\sum \hat D^2(S)|S| we derived Kesten’s bound var (D) =O(v).

2) We considered the value s so that \mu(D>s)=t, and showed by the basic inequality above that the variance of D conditioned on D>s is also bounded by v. This corresponds to exponential tail estimate proved by Kesten.

3) Using hypercontractivity we showed that the variance of D conditioned on D>s is actually bounded above by v/log (1/t) which corresponds to Talagrand’s sub-Gaussian tail-estimate.

4) Almost finally based on a certain very plausible lemma we used hypercontructivity to show that most Fourier coefficients of D are above the log v level, improving the variance upper bound to O(v/log v).

5) Since the plausible lemma is still open (see this MO question) we showed how we can “shortcut” the lemma and prove the upper bound without it.

The major open question

It is an open question to give an upper bound of v^{1-\epsilon} or even v^{2/3} which is the expected answer in dimension two. Michel Ledoux wisely proposes to prove it just for directed percolation in the plane (where all edges are directed up and right) from (0,0) to (v,v) where the edge length is Gaussian or Bernoulli.

Lecture 8

Three Further Applications of Discrete Fourier Analysis (without hypercontractivity)

The three next topics will use Fourier but not hypercontractivity. We start by talking about them.

1) The cap-set problem, some perspective and a little more extremal combinatorics

We talked about Roth theorem, the density Hales Jewett theorem,  the Erdos-Rado delta-system theorem and conjecture. We mentioned linearity testing.

2) Upper bounds for error-correcting codes

This was a good place to mention (and easily prove) a fundamental property used in both these cases:  The Fourier transform of convolutions of two functions f and g is the product of the Fourier transform of f and of g.

3) Social choice and Arrow’s theorem

The Fourier theoretic proof for Arrow’s theorem uses only Parseval’s formula so we are going to start with that.

Fourier-theoretic proof of Arrows theorem and related results.

We talked a little about Condorcet(we will later give a more detailed introduction to social choice). We mentioned Condorcet’s paradox, Condorcet’s Jury Theorem, and the notion of Condorcet winner.

Next we formulated Arrow’s theorem.  Lecture 9 was devoted to a Fourier-theoretic proof of Arrow theorem (in the balanced case). You can find it discussed in this blog post by Noam Nisan.  Lecture 10 mentioned a few further application of the Fourier method related to Arrow’s theorem, as well as a simple combinatorial proof of Arrow’s theorem in full generality. For the Fourier proof of Arrow’s theorem we showed that a Boolean function with all its non-zero Fourier coefficients on levels 0 and 1 is constant, dictatorship or anti-dictatorship. This time we formulated FKN theorem and showed how it implies a stability version of Arrow’s theorem in the neutral case.

Analysis of Boolean Functions – Week 3

Lecture 4

In the third week we moved directly to the course’s “punchline” – the use of Fourier-Walsh expansion of Boolean functions and the use of Hypercontractivity.

Before that we  started with  a very nice discrete isoperimetric question on a graph which is very much related to the graph of the discrete cube.

Consider the graph G whose vertices are 0-1 vectors of length n with precisely r ‘1’s, and with edges corresponding to vertices of Hamming distance two. (Which is the minimal Hamming distance between distinct vertices.) Given a set A of m vertices, how small can E(A, \bar A) be? (We already talked about intersecting families of sets of constant size – the Erdos-Ko-Rado theorem and, in general, it is a nice challenge to extend some of the ideas/methods of the course to constant weight situation.)

And now for the main part of the lecture.

1) Basics harmonic analysis on the discrete cube. We considered the vector space of real functions on the discrete cube \Omega_n and defined an inner product structure. We also defined the p-th norm for 1\le p\le \infty. Next we defined the Fourier-Walsh functions and showed that they form a orthonormal basis. This now leads to the Fourier-Walsh expansion for an arbitrary real function f on the discrete cube f=\sum_{S\subset[n]}\hat f(S)W_S, and we could easily verify Parseval formula.

2) Influence and Fourier. If f is a real function on \Omega_n and f=\sum \hat f(S)W_S its Fourier-Walsh expansion. We showed that I_k(f)=\sum_{S:k \in S}\hat f^2(S). It follows that I(f)=\sum_S\hat f^2(S)|S|. The Fourier-theoretic proof for I(f) ≥ 4 t (1-t) where t=μ(f) now follows easily.

3) Chinchine, hypercontractivity and the discrete isoperimetric inequality.

Next we discussed what will it take to prove the better estimate I(f) ≥ K t log t. We stated Chinchine inequality, and explained why is Chinchine inequality relevant: For Boolean functions the pth power of the p-norm does not depend on p. (It always equals t.) Therefore if t is small, the p-th norms themselves much be well apart! After spending a few moments on the history of the inequality (as told to me by Ron Blei) we discussed what kind of extension do we need, and stated  the Bonami-Gross-Beckner inequality. We use Bonami’s inequality to proof of the inequality I(f) ≥ K t log t and briefly talked about what more does it give.

Lecture 5

1) Review and examples. We reviewed what we did in the previous lecture and considered a few examples: Dictatiorship; the AND function (an opportunity to mention the uncertainty principle,) and MAJORITY on three variables. We also mentioned another connection between influences and Fourier-Walsh coefficients: for a monotone (non decreasing) Boolean function f, I_k(f) = -2\hat f(\{k\}).

2) KKL’s theorem

KKL’s theorem: There is an absolute constant K so that for every Boolean function f, with t=μ(f), there exists k, 1 ≤ k ≤ n such that

I_k(f) \ge K t (1-t) logn/n.

To prove of KKL’s theorem: we repeat, to a large extent, the steps from Lecture 4 (of course, the proof of KKL’s theorem was where this line of argument came from.) We showed that if all individual influences are below 1/sqrt n than I(f) \ge K t(1-t) \log n.

We mentioned one corollary: For Boolean function invariant under a transitive group of permutations, all individual influences are equal and therefore I(f) \ge K t (1-t)\log n.

3) Further problems

In the  last part of the lecture we mentioned seven problems regarding influence of variables and KKL’s theorem (and I added two here):

1) What can be said about balanced Boolean functions with small total influence?

2) What can be said about Boolean functions for which I(f) ≤ K t log (1/t), for some constant K, where t=μ(f)?

3) What can be said about the connection between the symmetry group and the minimum total influence?

4) What can be said about Boolean functions (1/3 ≤ μ(f)≤ 2/3, say) for which \max I_k(f) \le K log n/n.?

5) What more can be said about the vector of influences (I_1(f),I_2(f), \dots I_n(f))?

6)* What is the sharp constant in KKL’s theorem?

7)* What about edge expansions of (small) sets of vertices in general graphs?

8) Under what conditions I(f) \ge n^\beta for β >0.

9) What about influence of larger sets? In particular, what is the smallest t (as a function of n ) such that if \mu(f)=t there is a set S of variables S ≤ 0.3n with I_S(f) \ge 0.9?

(This post  is a short version, I will add details later on.)

Analysis of Boolean functions – week 2

Post on week 1; home page of the course analysis of Boolean functions

Lecture II:

We discussed two important examples that were introduced by Ben-Or and Linial: Recursive majority and  tribes.

Recursive majority (RM): F_m is a Boolean function with 3^m variables and F_{m+1} (x,y,z) = F_1(F_m(x),F_m(y),F_m(z)). For the base case we use the majority function F_1(x,y,z)=MAJ(x,y,z).

Tribes: Divide your n variables into s pairwise disjoint sets (“tribes”) of cardinality t. f=1 if for some tribe all variables equal one, and thus T=0 if for every tribe there is a variable with value ‘0’.

We note that this is not an odd function i.e. it is not symmetric with respect to switching ‘0’ and ‘1’. To have \mu=1/2 we need to set t=\log_2n - \log_2\log_2n+c. We computed the influence of every variable to be C log n/n. The tribe function is a depth-two formula of linear size and we briefly discussed what are Boolean formulas and Boolean circuits (These notions can be found in many places and also in this post.).

I states several conjectures and questions that Ben-Or and Linial raised in their 85 paper:

Conjecture 1: For every balanced Boolean function with n variables there is a variable k whose influence is \Omega (\log n/n).

Conjecture 2: For every balanced Boolean function with n variables there is a set S of n/log n variables whose influence I_S(f) is 1-o(1).

Question 3: To what extent can the bound in Conjecture 2 be improved if the function f is odd. (Namely, f(1-x_1,1-x_2,\dots, 1-x_n)=1-f(x_1,x_2,\dots, x_n).)

Our next theme was discrete isoperimetric results.  I noted the connection between total influence and edge expansion and proved the basic isoperimetric inequality: If μ(f)=t then I(f) ≥ 4 t(1-t). The proof uses the canonical paths argument.

Lecture III:

We proved using “compression” that sharp bound on I(f) as a function of t=μ(f). We made the analogy between compression and Steiner symmetrization – a classic method for proving the classical isoperimetric theorem. We discussed similar results on vertex boundary and on Talagrand-Margulis boundary (to be elaborated later in the course).

Then We proved the Harris-Kleitman inequality and showed how to deduce the fact that intersecting family of subsets of [n] with the property that the family of complements is also intersecting has at most 2^{n-2} sets.

The next topic is spectral graph theory. We proved the Hoffman bound for the largest size of an independent set in a graph G.

I mentioned graph-Laplacians and the spectral bound for expansions (Alon-Milman, Tanner)..

The proofs mentioned above are so lovely that I will add them on this page, but sometime later.

Next week I will introduce harmonic analysis on the discrete cube and give a Fourier-theoretic explanation for  the additional log (1/t) factor in the edge isoperimetric inequality.

Important announcement: Real analysis boot camp in the Simons Institute for the Theory of Computing, is part of the program in Real Analysis and Computer Science. It is taking place next week on September 9-13 and has three lecture series. All lecture series are related to the topic of the course and especially:

Poznań: Random Structures and Algorithms 2013

michal  photo

Michal Karonski (left) who built Poland’s probabilistic combinatorics group at Poznań, and a sculpture honoring the Polish mathematicians who first broke the Enigma machine (right, with David Conlon, picture taken by Jacob Fox).

I am visiting now Poznań for the 16th Conference on Random Structures and Algorithms. This bi-annually series of conferences started 30 years ago (as a satellite conference to the 1983 ICM which took place in Warsaw) and this time there was also a special celebration for Bela Bollobás 70th birthday. I was looking forward to  this first visit to Poland which is, of course, a moving experience for me. Before Poznań I spent a few days in Gdańsk visiting Robert Alicki. Today (Wednesday)  at the Poznań conference I gave a lecture on threshold phenomena and here are the slides. In the afternoon we had the traditional random run with a record number of runners. Let me briefly tell you about very few of the other lectures: Update (Thursday): A very good day, and among others a great talk of Jacob Fox on Relative Szemeredi Theorem (click for the slides from a similar talk from Budapest) where he presented a joint work with David Conlon and Yufei Zhao giving a very general and strong form of Szemeredi theorem for quasi-random sparse sets, which among other applications, leads to a much simpler proof of the Green -Tao theorem.

Mathias Schacht

Mathias Schacht gave a wonderful talk  on extremal results in random graphs (click for the slides) which describes some large recent body of highly successful research on the topic. Here are two crucial slides, and going through the whole presentation can give a very good overall picture. ms1 mt2

Vera Sós

Vera Sós gave an inspiring talk about the random nature of graphs which are extremal to the Ramsey property and connections with graph limits. Vera presented the following very interesting conjecture on graph limits. We say that a sequence of graphs (G_n) has a limit if for every k and every graph H with k vertices the proportion in G_n of induced H-subgraphs among all k-vertex induced subgraphs tend to a limit. Let us also say that (G_n) has a V-limit if for every k and every e the proportion in G_n of induced subgraphs with k vertices and e edges among all k-vertex induced subgraphs tend to a limit. Sós’ question: Is having a V-limit equivalent to having a limit. This is open even in the case of quasirandomness, namely, when the limit is given by the Erdos-Renyi model G(n,p). (Update: in this case V-limit is equivalent to limit, as several participants of the conference observed.) Both a positive and a negative answer to this fundamental question would lead to many further (different) open problems.

Joel Spencer

Joel Spencer gave a great (blackboard) talk about algorithmic aspects of the probabilistic method, and how existence theorems via the probabilistic method now often require complicated randomized algorithm. Joel mentioned his famous six standard deviation theorem. In this case, Joel conjectured thirty years ago that there is no efficient algorithm to find the coloring promised by his theorem. Joel was delighted to see his conjecture being refuted first by Nikhil Bansal (who found an algorithm whose proof depends on the theorem) and then later by Shachar Lovett and  Raghu Meka (who found a new algorithm giving a new proof) . In fact, Joel said, having his conjecture disproved is even more delightful than having it proved. Based on this experience Joel and I are now proposing another conjecture: Kalai-Spencer (pre)conjecture: Every existence statement proved by the probabilistic method can be complemented by an efficient (possibly randomized) algorithm. By “complemented by an efficient algorithm” we mean that there is an efficient(polynomial time)  randomized algorithm to create the promised object with high probability.  We refer to it as a preconjecture since the term “the probabilistic method” is not entirely well-defined. But it may be possible to put this conjecture on formal grounds, and to discuss it informally even before.

BosonSampling and (BKS) Noise Sensitivity

Update (Nov 2014): Noise sensitivity of BosonSampling and computational complexity of noisy BosonSampling are studied in this paper by Guy Kindler and me. Some of my predictions from this post turned out to be false. In particular the noisy BosonSampling is not  flat and it does depend on the input matrix.  However when the noise level is a constant BosonSampling is in P, and when it is above 1 over the number of bosons, we cannot expect robust experimental outcomes.

 

—–

 

Following are some preliminary observations connecting BosonSampling, an interesting  computational task that quantum computers can perform (that we discussed in this post), and noise-sensitivity in the sense of Benjamini, Schramm, and myself (that we discussed here and here.)

BosonSampling and computational-complexity hierarchy-collapse

Suppose that you start with n bosons each can have m locations. The i-th boson is in superposition and occupies the j-th location with complex weight a_{ij}. The bosons are indistinguishable which makes the weight for a certain occupation pattern proportional to the permanent of a certain n by n submatrix of the n by m matrix of weights.

Boson Sampling is a task that a quantum computer can perform. As a matter of fact, it only requires a “boson machine” which represents only a fragment of quantum computation. A boson machine is a quantum computer which only manipulates indistinguishable bosons with gated described by phaseshifters and beamsplitters.

BosonSampling and boson machines were studied in a recent paper The Computational Complexity of Linear Optics of Scott Aaronson and Alex Arkhipov (AA). They proved (Theorem 1 in the paper) that if (exact) BosonSampling can be performed by a classical computer then this implies a collapse of the computational-complexity polynomial hierarchy (PH, for short). This result adds to a similar result achieved independently by Michael J. Bremner, Richard Jozsa, and Dan J. Shepherd (BJS) in the paper entitled: “Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy,” and to older results by  Barbara Terhal and David DiVincenzo (TD) in the paper Adaptive quantum computation, constant depth quantum circuits and Arthur-Merlin games, Quant. Inf. Comp. 4, 134-145 (2004).

Since universal quantum computers can achieve BosonSampling (and the other related computational tasks considered by TD and BJS), this is a very strong indication for the computational complexity advantage of quantum computers which arguably brings us with quantum computers to the “cozy neighborhood” of NP-hardness.

Noisy quantum computers with quantum fault-tolerance are also capable of exact BosonSampling and this strong computational-complexity quantum-superiority applies to them as well.

Realistic BosonSampling and Gaussian Permanent Estimation (GPE)

Aaronson an Arkhipov considered the following question that they referred to as Gaussian Permanent Approximation.

Problem (Problem 2 from AA’s paper): (|GPE|_{\pm}^2): Given as imput a matrix {\cal N}(0,1)_{\bf C}^{n \times n} of i.i.d Gaussians,together with error bounds ε, δ > o, estimate to within additive error \pm \epsilon n! with probability at leat 1-δ over X, in poly(n,1/\epsilon,1/\delta) time.

They conjectured that such Gaussian Permanent Approximation is computationally hard and showed (Theorem 3) that this would imply that sampling w.r.t. states achievable by boson machines cannot even be approximated by classical computing (unless PH collapses). They regarded questions about approximation more realistic in the context of decoherence where we cannot expect exact sampling.

Scott Aaronson also expressed guarded optimism that even without quantum fault-tolerance BosonSampling can be demonstrated by boson machines for 20-30 bosons, leading to strong experimental evidence for computational advantage of quantum computers (or, if you wish, against the efficient Church-Turing thesis).

Is it so?

More realistic BosonSampling and Noisy Gaussian Permanent Estimation (NGPE)

Let us consider the following variation that we refer to as Noisy Gaussian Permanent Estimation:

Problem 2′: (|NGPE|_{\pm}^2): Given as imput a matrix M= {\cal N}(0,1)_{\bf C}^{n \times n} of i.i.d Gaussians, and a parameter t>0 let NPER (M),  be the expected value of the permanent for \sqrt {1-t^2}M+E where E= {\cal N}(0,t)_{\bf C}^{n \times n}.  Given the input matrix M together with error bounds ε, δ > o, estimate NPER(M) to within additive error \pm \epsilon n! with probability at leat 1-δ over X, in poly(n,1/\epsilon,1/\delta) time.

Problem 2′ seems more relevant for noisy boson machines (without fault-tolerance). The noisy state of the computer is based on every single boson  being slightly mixed, and the permanent computation will average these individual mixtures. We can consider the relevant value for t to be a small constant. Can we expect Problem 2′ to be hard?

The answer for Question 2′ is surprising. I expect that even when t is very very tiny, namely t=n^{-\beta} for \beta <1, the expected value of NPER(M) (essentially) does not depend at all on M!

Noise Sensitivity a la Benjamini, Kalai and Schramm

Noise sensitivity for the sense described here for Boolean functions was studied in a paper by Benjamini Schramm and me in 1999.  (A related notion was studied by Tsirelson and Vershik.) Lectures on noise sensitivity and percolation is a new beautiful monograph by Christophe Garban and Jeff Steif which contains a description of noise sensitivity. The setting extends easily to the Gaussian case. See this paper by Kindler and O’donnell for the Gaussian case. In 2007, Ofer Zeituni and I studied the noise sensitivity in the Gaussian model of the maximal eigenvalue of random Gaussian matrices (but did not write it up).

NS

Noise sensitivity depends on the degree of the support of the Fourier expansion. For determinants or permanents of an n by n matrices the basic formulas as sums of generalized diagonals describe the Fourier expansion,  so the Fourier coefficients are supported on degree-n monomials. This implies that the determinant and the permanent are very noise sensitive.

Noisy Gaussian Permanent Estimation is easy

Noisy Gaussian Permanent Estimation is easy, even for very small amount of noise, because the outcome does not depend at all on the input. It is an interesting question what is the hardness of NGPE is when the noise is below the level of noise sensitivity.

Update (March, 2014) Exploring the connection between BosonSampling and BKS-noise sensitivity shows that the picture drawn here is incorrect. Indeed, the square of the permanent is not noise stable even when the noise is fairly small and this shows that the noisy distribution does not approximate the noiseless distribution. However the noisy distribution does depend on the input!

 

AA’s protocol and experimental BosonSampling

Scott and Alex proposed a simple experiment described as follows : “An important motivation for our results is that they immediately suggest a linear-optics experiment, which would use simple optical elements (beamsplitters and phaseshifters) to induce a Haar-random m \times m unitary transformation U on an input state of n photons, and would then check that the probabilities of various final states of the photons correspond to the permanents of n \times n submatrices, as predicted by quantum mechanics.”

Recently, four groups carried out interesting BosonSampling experiments with 3 bosons (thus for permanents of 3 by 3 matrices.) (See this post on Scott’s blog.)

BKS-noise sensitivity is giving  simple predictions on how things will behave as a function of the number of bosons and this can be tested even with experiments with very small number of bosons. When you increase the number of bosons the distribution will quickly become uniform for the various final states. The correlation between the probabilities and the value corresponding to permanents will rapidly go to zero.

Some follow-up questions

Here are a few interesting questions that deserve further study.

1) Does problem 2′ capture the general behavior of noisy boson machines? To what generality noise sensitivity applies for general functions described by Boson sampling distributions?

(There are several versions for photons-based quantum computers including even an important  model by Knill, Laflamme, and Milburn that support universal quantum computation. The relevance of BKS noise-sensitivity should be explored carefully for the various versions.)

2) Is the connection with noise sensitivity relevant to the possibility to have boson machines with fault tolerance?

3) What is the Gaussian-quantum analog for BKS’s theorem asserting that noise sensitivity is the law unless  we have substantial correlation with the majority function?

4) What can be said about noise-sensitivity of measurements for other quantum codes?

A few more remarks:

More regarding noisy boson machines and quantum fault tolerance

Noisy boson machines and BosonSampling are related to various other issues regarding quantum fault-tolerance. See my added recent remarks (about the issue of synchronization, and possible modeling using smoothed Lindblad evolutions) to my old post on AA’s work.

Noise sensitivity and the special role of the majority function

bks

The main result of Itai, Oded, and me was that a Boolean function which is not noise sensitive must have a substantial correlation with the majority function. Noise sensitivity and the special role of majority for it gave me some motivation to look at quantum fault-tolerance in 2005  (see also this post,) and this is briefly discussed in my first paper on the subject, but until now I didn’t find an actual connection between quantum fault-tolerance and BKS-noise-sensitivity.

Censorship

It is an interesting question which bosonic states are realistic, and it came up in some of my papers and in the discussion with Aram Harrow and Steve Flammia (and their paper on my “Conjecture C”).

A sort of conclusion

BosonSampling was offered as a way to prove that quantum mechanics allows a computational advantage without using the computational advantage for actual algorithmic purpose. If you wish, the AA’s protocol is offered as a sort of zero-knowledge proof that the efficient Church-Turing thesis is false.  It is a beautiful idea that attracted interest and good subsequent work from theoreticians and experimentalists. If the ideas described here are correct, BosonSampling and boson machines may give a clear understanding based on BKS noise-sensitivity for why quantum mechanics does not offer computational superiority (at least not without the magic of quantum fault-tolerance).

Avi’s joke and common sense

Here is a quote from AA referring to a joke by Avi Wigderson: “Besides bosons, the other basic particles in the universe are fermions; these include matter particles such as quarks and electrons. Remarkably, the amplitudes for n-fermion processes are given not by permanents but by determinants of n×n matrices. Despite the similarity of their definitions, it is well-known that the permanent and determinant differ dramatically in their computational properties; the former is #P-complete while the latter is in P. In a lecture in 2000, Wigderson called attention to this striking connection between the boson and fermion dichotomy of physics and the permanent-determinant dichotomy of computer science. He joked that, between bosons and fermions, ‘the bosons got the harder job.’ One could view this paper as a formalization of Wigderson’s joke.”

While sharing the admiration to Avi in general and Avi’s jokes in particular, if we do want to take Avi’s joke seriously (as we always should), then the common-sense approach would be first to try to understand why is it that nature treats bosons and fermions quite equally and the dramatic computational distinction is not manifested at all. The answer is that a crucial ingredient for a computational model is the modeling of noise/errors, and that noise-sensitivity makes bosons and fermions quite similar physically and computationally.

Eigenvalues, determinants, and Yuval Filmus

It is an interesting question (that I asked over Mathoverflow) to understand the Fourier expansion of the set of eigenvalues, the maximum eigenvalue and related functions. At a later point,  last May,  I was curious about the Fourier expansion of the determinant, and for the Boolean case I noticed remarkable properties of its Fourier expansion. So I decided to ask Yuval Filmus about it:

Dear Yuval

 I am curious about the following. Let D be the function defined on {-1,1}^n^2
which associates to every +1/1 matrix its determinant.
What can be said about the Fourier transform of D? It looks to me that easy arguments shows that the Fourier transform is supported only on subsets of the entries
so that in every raw and columns there are odd number of entries. Likely there are even further restrictions that I would be interested to explore.
Do you know anything about it?
best Gil

Yuval’s answer came a couple of hours later like a cold shower:

Hi Gil,

The determinant is a sum of products of generalized diagonals.
Each generalized diagonal is just a Fourier character, and they are all different.

In other words, the usual formula for the determinant *is* its Fourier transform

This reminded me of a lovely story of how I introduced Moni Naor to himself that I should tell sometime.

What else can a quantum computer sample?

The ability of quantum computers to sample (exactly) random complex Gaussian matrices according to the value of their permanents is truly amazing! If you are not too impressed by efficient factoring but still do not believe that QC can reach the neighborhood of NP-hard problems you may find this possibility too good to be true.

I am curious if sharp P reductions give us further results of this nature. For example,  can a QC sample random 3-SAT formulas (by a uniform distribution or by a certain other distribution coming from sharp-P reductions) according to the number of their satisfying assignments?

Can QC sample integer polytopes by their volume or by the number of integer points in them? Graphs by the number of 4-colorings?

Lawler-Kozdron-Richards-Stroock’s combined Proof for the Matrix-Tree theorem and Wilson’s Theorem

wilson  curvature

David Wilson and a cover of Shlomo’s recent book “Curvature in mathematics and physics”

A few weeks ago, in David Kazhdan’s basic notion seminar, Shlomo Sternberg gave a lovely presentation Kirchho ff and Wilson via Kozdron and Stroock. The lecture is based on the work presented in the very recent paper by Michael J. Kozdron,  Larissa M. Richards, and Daniel W. Stroock: Determinants, their applications to Markov processes, and a random walk proof of Kirchhoff’s matrix tree theorem. Preprint, 2013. Available online at arXiv:1306.2059.

Here is the abstract:

Kirchhoff’s formula for the number of spanning trees in a connected graph  is over 150 years old. For example, it says that if c_2, \dots, c_n are the nonzero  eigenvalues of the Laplacian, then the number k of spanning trees is k= (1/n)c_2\cdots c_n. There are many proofs.  An algorithm due to Wilson via loop erased random walks produces such a tree, and Wilson’s theorem is that all spanning trees are produced by his algorithm with equal probability. Hence,  after the fact, we know that Wilson’s algorithm produces any given tree with probability 1/k.  A proof due to Lawler, using the Green’s function, shows directly that Wilson’s algorithm has the probability 1/k  of producing any given spanning tree, thus simultaneously proving Wilson’s theorem and Kirchhoff’s formula. Lawler’s proof has been considerably simplified by Kozdron and Stroock. I plan to explain their proof. The lecture will be completely self-contained, using only Cramer’s rule from linear algebra.

(Here are also lecture notes of the lecture by Ron Rosenthal.)

Here is some background.

The matrix-tree theorem

The matrix tree theorem asserts that the number of rooted spanning trees of a connected graph G  is the product of the non-zero eigenvalues of L(G), the Laplacian of G.

Suppose that G has n vertices. The Laplacian of G is the matrix whose (i,i)-entry is the degree of the ith vertex, and its (i,j) entry for i \ne j is 0 if the ith vertex is not adjacent to the jth vertex, and -1 if they are adjacent. So  L(G)=D-A(G) where A(G) is the adjacency matrix of G, and D is a diagonal matrix whose entries are the degrees of the vertices.  An equivalent formulation of the matrix-tree theorem is that the number of spanning trees is the determinant of a matrix obtained from the Laplacian by deleting the j th row and j th column.

We considered a high dimensional generalization of the matrix tree theorem in these posts (I, II, III, IV).

How to generate a random spanning tree for a graph G?

Using the matrix-tree theorem

Method A: Start with an edge e \in G, use the matrix-tree theorem to compute the probability p_e that e belongs to a random spanning tree of G, take e with probability p_e. If e is taken consider the contraction G/e and if G is not taken consider the deletion G \backslash e and continue.

This is an efficient method to generate a random spanning tree according to the uniform probability distribution. You can extend it by assigning each edge a weight and chosing a tree with probability proportional to the product of its weights.

Random weights and greedy

Method B: Assign each edge a random real number between 0 and 1 and chose the spanning tree which minimizes the sum of weights via the greedy algorithm.

This is a wonderful method but it leads to a different probability distribution on random spanning trees which is very interesting!

The Aldous-Broder random walk method

Method C: The Aldous-Broder theorem. Start a simple random walk from a vertex of the graph until reaching all vertices, and take each edge that did not form a cycle with earlier edges. (Or, in other words, take every edge that reduced the number of connected components of the graph on the whole vertex set and visited edges.)

Amazingly, this leads to a random uniform spanning tree. The next method is also very amazing and important for many applications.

David Wilson’s algorithm

Method D: Wilson’s algorithm. Fix a vertex as a root. (Later the root will be a whole set of vertices, and a tree on them.) Start from an arbitrary vertex u not in the root and take a simple random walk until you reach the root. Next, erase all edges in cycles of the path created by the random walk so you will left with a simple path from  u to the root. Add this path to the root and continue!

Here is a link to Wilson’s paper! Here is a nice presentation by Chatterji  and Gulwani.

Oz’ Balls Problem: The Solution

idledisc500

A commentator named Oz proposed the following question: You have a box with n red balls and n blue balls. You take out each time a ball at random but, if the ball was red, you put it back in the box and take out a blue ball. If the ball was blue, you put it back in the box and take out a red ball.

You keep doing it until left only with balls of the same color. How many balls will be left (as a function of n)?

ozpoll

Peter Shor wrote in a comment “I’m fairly sure that there is not enough bias to get cn, but it intuitively seems far too much bias to still be c \sqrt{n}. I want to say n^c. At a wild guess, it’s either c = \frac{2}{3}or c = \frac{3}{4}, since those are the simplest exponents between \frac{1}{2} and 1.”  The comment followed by a heuristic argument of Kevin Kostelo and computer experiments by Lior Silberman that supported the answer n^{3/4}.
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Taking balls away: Oz’ Version

This post is based on a comment by Oz to our question about balls with two colors:

“There is an interesting (and more difficult) variation I once heard but can’t recall where:

You have a box with n red balls and n blue balls. You take out each time a ball at random as before. But, if the ball was red, you put it back in the box and take out a blue ball. If the ball was blue, you put it back in the box and take out a red ball.

You keep as before until left only with balls of the same color. How many balls will be left (as a function of n)?

1) Roughly  εn for some ε>0.

2) Roughly \sqrt n?

3) Roughly log n?

4) Roughly a constant?

5) Some other behavior