Matrix rigidity, and all that

The rigidity challenge asks to exhibit an n × n matrix M that cannot be written as M = A + B where A is “sparse” and B is “low-rank.” This challenge was raised by Valiant who showed in [Val77] that if it is met for any A with at most n1+ϵ non-zero entries and any B with rank O(n∕ log log n) then computing the linear transformation M requires either logarithmic depth or superlinear size for linear circuits. This connection relies on the following lemma.


Lemma 1. Let C : {0, 1}n →{0, 1}n be a circuit made of XOR gates. If you can remove e edges and reduce the depth to d then the linear transformation computed by C equals A + B where A has ≤ 2d non-zero entries per row (and so a total of ≤ n2d non-zero entries), and B has rank ≤ e.


Proof: After you remove the edges, each output bit is a linear combination of the removed edges and at most 2d input variables. The former can be done by B, the latter by A. QED

Valiant shows that in a log-depth, linear-size circuit one can remove O(n∕ log log n) edges to reduce the depth to nϵ – a proof can be found in [Vio09] – and this gives the above connection to lower bounds.

However, the best available tradeoff for explicit matrices give sparsity n2∕r log(n∕r) and rank r, for any parameter r; and this is not sufficient for application to lower bounds.

Error-correcting codes

It was asked whether generator matrixes of good linear codes are rigid. (A code is good if it has constant rate and constant relative distance. The dimensions of the corresponding matrixes are off by only a constant factor, and so we can treat them as identical.) Spielman [Spi95] shows that there exist good codes that can be encoded by linear-size logarithmic depth circuits. This immediately rules out the possibility of proving a lower bound, and it gives a non-trivial rigidity upper bound via the above connections.

Still, one can ask if these matrices at least are more rigid than the available tradeoffs. Goldreich reports a negative answer by Dvir, showing that there exist good codes whose generating matrix C equals A + B where A has at most O(n2∕d) non-zero entries and B has rank O(d log n∕d), for any d.

A similar negative answer follows by the paper [GHK+13]. There we show that there exist good linear codes whose generating matrix can be written as the product of few sparse matrixes. The corresponding circuits are very structured, and so perhaps it is not surprising that they give good rigidity upper bounds. More precisely, the paper shows that we can encode an n-bit message by a circuit made of XOR gates and with say n log *n wires and depth O(1) – with unbounded fan-in. Each gate in the circuit computes the XOR of some t gates, which can be written as a binary tree of depth log 2t + O(1). Such trees have poor rigidity:


Lemma 2.[Trees are not rigid] Let C be a binary tree of depth d. You can remove an O(1∕2b) fraction of edges to reduce the depth to b, for any b.


Proof: It suffices to remove all edges at depths d – b, d – 2b, …. The number of such edges is O(2d-b + 2d-2b + …) = O(2d-b). Note this includes the case d ≤ b, where we can remove 0 edges. QED

Applying Lemma 2 to a gate in our circuit, we reduce the depth of the binary tree computed at that gate to b. Applying this to every gate we obtain a circuit of depth O(b). In total we have removed an O(1∕2b) fraction of the n log *n edges.

Writing 2b = n∕d, by Lemma 1 we can write the generating matrixes of our code as C = A + B where A has at most O(n∕d) non-zero entries per row, and B has rank O(d log *n). These parameters are the same as in Dvir’s result, up to lower-order terms. The lower-order terms appear incomparable.

Walsh-Fourier transform

Another matrix that was considered is the n×n Inner Product matrix H, aka the Walsh-Hadamard matrix, where the x,y entry is the inner product of x and y modulo 2. Alman and Williams [AW16] recently give an interesting rigidity upper bound which prevents this machinery to establish a circuit lower bound. Specifically they show that H can be written as H = A + B where A has at most n1+ϵ non-zero entries, and B has rank n1-ϵ′, for any ϵ and an ϵ′ which goes to 0 when ϵ does.

Their upper bound works as follows. Let h = log 2n. Start with the univariate, real polynomial p(z1,z2,…,zh) which computes parity exactly on inputs of Hamming weight between 2ϵn and (1∕2 + ϵ)n. By interpolation such a polynomial exists with degree (1∕2 – ϵ)n. Replacing zi with xiyi you obtain a polynomial of degree n – ϵn which computes IP correctly on inputs x,y whose inner product is between 2ϵn and (1∕2 + ϵ)n.

This polynomial has 2(1-ϵ′)n monomials, where ϵ′ = Ω(ϵ2). The truth-table of a polynomial with m monomials is a matrix with rank m, and this gives a low-rank matrix B′.

The fact that sparse polynomials yield low-rank matrixes also appeared in the paper [SV12], which suggested to study the rigidity challenge for matrixes arising from polynomials.

Returning to the proof in [AW16], it remains to deal with inputs whose inner product does not lie in that range. The number of x whose weight is not between (1∕2 – ϵ)n and (1∕2 + ϵ)n is 2(1-ϵ′)n. For each such input x we modify a row of the matrix B′. Repeating the process for the y we obtain the matrix B, and the rank bound 2(1-ϵ′)n hasn’t changed.

Now a calculation shows that B differs from H in few entries. That is, there are few x and y with Hamming weight between (1∕2 – ϵ)n and (1∕2 + ϵ)n, but with inner product less than 2ϵn.

Boolean complexity

There exists a corresponding framework for boolean circuits (as opposed to circuits with XOR gates only). Rigid matrixes informally correspond to depth-3 Or-And-Or circuits. If this circuit has fan-in fo at the output gate and fan-in fi at each input gate, then the correspondence in parameters is

rank = log fo
sparsity = 2fi .

More precisely, we have the following lemma.


Lemma 3. Let C : {0, 1}n →{0, 1}n be a boolean circuit. If you can remove e edges and reduce the depth to d then you can write C as an Or-And-Or circuit with output fan-in 2e and input fan-in 2d.


Proof: After you remove the edges, each output bit and each removed edge depends on at most 2d input bits or removed edges. The output Or gate of the depth-3 circuit is a big Or over all 2e assignments of values for the removed edges. Then we need to check consistency. Each consistency check just depends on 2d inputs and so can be written as a depth-2 circuit with fan-in 2d. QED

The available bounds are of the form log fo = n∕fi. For example, for input fan-in fi = nα we have lower bounds exponential in n1-α but not more. Again it can be shown that breaking this tradeoff in certain regimes (namely, log 2fo = O(n∕ log log n)) yields lower bounds against linear-size log-depth circuits. (A proof appears in [Vio09].) It was also pointed out in [Vio13] that breaking this tradeoff in any regime yields lower bounds for branching programs. See also the previous post.

One may ask how pairwise independent hash functions relate to this challenge. Ishai, Kushilevitz, Ostrovsky, and Sahai showed [IKOS08] that they can be computed by linear-size log-depth circuits. Again this gives a non-trivial upper bound for depth-3 circuits via these connections, and one can ask for more. In [GHK+13] we give constructions of such circuits which in combination with Lemma 3 can again be used to almost match the available trade-offs.

The bottom line of this post is that we can’t prove lower bounds because they are false, and it is a puzzle to me why some people appear confident that P is different from NP.

References

[AW16]    Josh Alman and Ryan Williams. Probabilistic rank and matrix rigidity, 2016. https://arxiv.org/abs/1611.05558.

[GHK+13]   Anna Gál, Kristoffer Arnsfelt Hansen, Michal Koucký, Pavel Pudlák, and Emanuele Viola. Tight bounds on computing error-correcting codes by bounded-depth circuits with arbitrary gates. IEEE Transactions on Information Theory, 59(10):6611–6627, 2013.

[IKOS08]    Yuval Ishai, Eyal Kushilevitz, Rafail Ostrovsky, and Amit Sahai. Cryptography with constant computational overhead. In 40th ACM Symp. on the Theory of Computing (STOC), pages 433–442, 2008.

[Spi95]    Daniel Spielman. Computationally Efficient Error-Correcting Codes and Holographic Proofs. PhD thesis, Massachusetts Institute of Technology, 1995.

[SV12]    Rocco A. Servedio and Emanuele Viola. On a special case of rigidity. Available at http://www.ccs.neu.edu/home/viola/, 2012.

[Val77]    Leslie G. Valiant. Graph-theoretic arguments in low-level complexity. In 6th Symposium on Mathematical Foundations of Computer Science, volume 53 of Lecture Notes in Computer Science, pages 162–176. Springer, 1977.

[Vio09]    Emanuele Viola. On the power of small-depth computation. Foundations and Trends in Theoretical Computer Science, 5(1):1–72, 2009.

[Vio13]    Emanuele Viola. Challenges in computational lower bounds. Available at http://www.ccs.neu.edu/home/viola/, 2013.

Mixing in groups, II

In the previous post we have reduced the “three-step mixing” over SL(2,q), the group of 2×2 matrices over the field with q elements with determinant 1, to the following statement about mixing of conjugacy classes.


Theorem 1.[Mixing of conjugacy classes of SL(2,q)] Let G = SL(2,q). With probability ≥ 1 -|G|-Ω(1) over uniform a,b in G, the distribution C(a)C(b) is |G|-Ω(1) close in statistical distance to uniform.


Here and throughout this post, C(g) denotes a uniform element from the conjugacy class of g, and every occurrence of C corresponds to an independent draw.

In this post we sketch a proof of Theorem 1, following [GV15]. Similar theorems were proved already. For example Shalev [Sha08] proves a version of Theorem 1 without a quantitative bound on the statistical distance. It is possible to plug some more representation-theoretic information into Shalev’s proof and get the same quantitative bound as in Theorem 1, though I don’t have a good reference for this extra information. However the proof in [Sha08] builds on a number of other things, which also means that if I have to prove a similar but not identical statement, as we had to do in [GV15], it would be hard for me.

Instead, here is how you can proceed to prove the theorem. First, we remark that the distribution of C(a)C(b) is the same as that of

C(C(a)C(b)),

because for uniform x, y, and z in Fq we have the following equalities of distributions:

C(C(a)C(b)) = x-1(y-1ayz-1bz)x = x-1(y-1ayxx-1z-1bz)x = C(a)C(b)

where the last equality follows by replacing y with yx and z with zx.

That means that we get one conjugation “for free” and we just have to show that C(a)C(b) falls into various conjugacy classes with the right probability.

Now the great thing about SL(2,q) is that you can essentially think of it as made up of q conjugacy classes each of size q2 (the whole group has size q3 – q). This is of course not exactly correct, in particular the identity element obviously gives a conjugacy class of size 1. But except for a constant number of conjugacy classes, every conjugacy class has size q2 up to lower-order terms. This means that what we have to show is simply that the conjugacy class of C(a)C(b) is essentially uniform over conjugacy classes.

Next, the trace map Tr  : SL(2,q) → Fq is essentially a bijection between conjugacy classes and the field Fq. To see this recall that the trace map satisfies the cyclic property:

Tr xyz = Tr yzx.

This implies that

Tr u-1au = Tr auu-1 = Tr a,

and so conjugate elements have the same trace. On the other hand, the q matrixes

x  1

1  0

for x in Fq all have different traces, and by what we said above their conjugacy classes make up essentially all the group.

Putting altogether, what we are trying to show is that

Tr C(a)C(b)

is |G|-Ω(1) close to uniform over F q in statistical distance.

Furthermore, again by the cyclic property we can consider without loss of generality

Tr aC(b)

instead, and moreover we can let a have the form

0  1

1  w

and b have the form

v  1

1  0

(there is no important reason why w is at the bottom rather than at the top).

Writing a generic g in SL(2,q) as the matrix

u1   u2

u3   u4

you can now with some patience work out the expression

Tr au-1bu = vu 3u4 – u32 + u 42 – vu 1u2 + u12 – vwu 2u3 + wu1u3 – u22 – wu 2u4.

What we want to show is that for typical choices of w and v, the value of this polynomial is q-Ω(1) close to uniform over F q for a uniform choice of u subject to the determinant of u being 1, i.e, u1u4 – u2u3 = 1.

Maybe there is some machinery that immediately does that. Lacking the machinery, you can use the equation u1u4 – u2u3 = 1 to remove u4 by dividing by u1 (the cases where u1 = 0 are few and do not affect the final answer). Now you end up with a polynomial p in three variables, which we can rename x, y, and z. You want to show that p(x,y,z) is close to uniform, for uniform choices for x,y,z. The benefit of this substitution is that we removed the annoying condition that the determinant is one.

To argue about p(x,y,z), the DeMillo–Lipton–Schwartz-Zippel lemma comes to mind, but is not sufficient for our purposes. It is consistent with that lemma that the polynomial doesn’t take a constant fraction of the values of the field, which would give a constant statistical distance. One has to use more powerful results known as the Lang-Weil theorem. This theorem provides under suitable conditions on p a sharp bound on the probability that p(x,y,z) = a for a fixed a in Fq. The probability is 1∕q plus lower-order terms, and then by summing over all a in Fq one obtains the desired bound on the statistical distance.

I am curious if there is a way to get the statistical distance bound without first proving a point-wise bound.

To apply the Lang-Weil theorem you have to show that the polynomial is “absolutely irreducible,” i.e., irreducible over any algebraic extension of the field. This can be proven from first principles by a somewhat lengthy case analysis.

References

[GV15]   W. T. Gowers and Emanuele Viola. The communication complexity of interleaved group products. In ACM Symp. on the Theory of Computing (STOC), 2015.

[Sha08]   Aner Shalev. Mixing and generation in simple groups. J. Algebra, 319(7):3075–3086, 2008.

Mixing in groups

Non-abelian groups behave in ways that are useful in computer science. Barrington’s famous result [Bar89] shows that we can write efficiently an arbitrary low-depth computation as a group product over any non-solvable group. (Being non-solvable is a certain strengthening of being non-abelian which is not important now.) His result, which is false for abelian groups, has found myriad applications in computer science. And it is amusing to note that actually results about representing computation as group products were obtained twenty years before Barrington, see [KMR66]; but the time was not yet ripe.

This post is about a different property that certain non-abelian groups have and that is also useful. Basically, these groups ”mix” in the sense that if you have several distributions over the group, and the distributions have high entropy, then the product distribution (i.e., sample from each distribution and output the product) is very close to uniform.

First, let us quickly remark that this is completely false for abelian groups. To make an example that is familiar to computer scientists, consider the group of n-bit strings with bit-wise xor. Now let A be the uniform distribution over this group where the first bit is always 0. Then no matter how many independent copies of A you multiply together, the product is always A.

Remarkably, over other groups it is possible to show that the product distribution will become closer and closer to uniform. A group that works very well in this respect is SL(2,q), the group of 2×2 matrices over the field with q elements with determinant 1. This is a group that in some sense is very far from abelian. In particular, one can prove the following result.


Theorem 1.[Three-step mixing [Gow08BNP08]] Let G = SL(2,q), and let A, B, and C be three subsets of G of constant density. Let a, b, and c be picked independently and uniformly from A, B, and C respectively. Then for any g in G we have

| Pr[abc = g] – 1∕|G|| < 1∕|G|1+Ω(1).


Note that the conclusion of the theorem in particular implies that abc is supported over the entire group. This is remarkable, since the starting distributions are supported over only a small fraction of the group. Moreover, by summing over all elements g in the group we obtain that abc is polynomially close to uniform in statistical distance.

Theorem 1 can be proved using representation theory. This must be a great tool, but for some reason I always found it a little difficult to digest the barrage of definitions that usually anticipate the interesting stuff.

Luckily, there is another way to prove Theorem 1. I wouldn’t be surprised if this is in some sense the same way, and moreover this other way is not sometimes I would call elementary. But it is true that I will be able to sketch a proof of the theorem without using the word ”representation”. In this post we will prove some preliminary results that are valid for all groups, and the most complicated thing used is the Cauchy-Schwarz inequality. In the next post we will work specifically with the group SL(2,q), and use more machinery. This is all taken from this paper with Gowers [GV15] (whose main focus is the study of mixing in the presence of dependencies).

First, for convenience let us identify a set A with its characteristic function. So we write A(a) for a belongs to the set A. It is convenient to work with a slightly different statement:


Theorem 2. Let G = SL(2,q) and let A,B,C be three subsets of G of densities α,β,γ respectively. For any g in G,

|Eabc=gA(a)B(b)C(c) – αβγ|≤|G|-Ω(1)

where the expectation is over uniform elements a, b, and c from the group G such that their product is equal to g.


This Theorem 2 is equivalent to Theorem 1, because

Eabc=gA(a)B(b)C(c) = Pr[A(a),B(b),C(c)|abc = g]
= Pr[abc = g|A(a),B(b),C(c)]|G|αβγ

by Bayes’ rule. So we can get Theorem 1 by dividing by |G|αβγ.

Now we observe that to prove this ”mixing in three steps” it actually suffices to prove mixing in four steps.


Theorem 3.[Mixing in four steps] Let G = SL(2,q) and let A,B,C,D be four subsets of G of densities α,β,γ,δ respectively. For any g in G,

Eabcd=gA(a)B(b)C(c)D(d) – αβγδ ≤|G|-Ω(1),

where the expectation is over uniform elements a, b, c, and d from the group G such that their product is equal to g.



Lemma 4. Mixing in four steps implies mixing in three.


Proof: Rewrite

|Eabc=gA(a)B(b)C(c) – αβγ| = |Eabc=gf(a)B(b)C(c)|

where f(a) := A(a) – α.

In these proofs we will apply Cauchy-Schwarz several times. Each application ”loses a square,” but since we are aiming for an upper bound of the form 1∕|G|Ω(1) we can afford any constant number of applications. Our first one is now:

(Eabc=gf(a)B(b)C(c))2 ≤ (E cC(c)2)(E c(Eab=gc-1f(a)B(b))2)
= γEcEab=a′b′=gc-1f(a)B(b)f(a′)B(b′)
= γEab=a′b′(A(a) – α)B(b)B(b′)(A(a′) – α).

There are four terms that make up the expectation. The terms that involve at least one α sum to -α2β2. The remaining term is the expectation of A(a)B(b)B(b′)A(a′). Note that ab = a′b′ is equivalent to ab(1∕b′)(1∕a′) = 1G. Hence by Theorem 3 this expectation is at most |G|-Ω(1). QED

So what remains to see is how to prove mixing in four steps. We shall reduce the mixing problem to the following statement about the mixing of conjugacy classes of our group.


Definition 5. We denote by C(g) the conjugacy class {h-1gh : h in G} of an element g in G. We also denote by C(g) the uniform distribution over C(g) for a uniformly selected g in G.



Theorem 6.[Mixing of conjugacy classes of SL(2,q)] Let G = SL(2,q). With probability ≥ 1 -|G|-Ω(1) over uniform a,b in G, the distribution C(a)C(b) is |G|-Ω(1) close in statistical distance to uniform.


Theorem 6 is proved in the next blog post. Here we just show that is suffices for our needs.


Lemma 7. Theorem 6 implies Theorem 3.


Proof: We rewrite the quantity to bound as

Ea,c(A(a)C(c)Eb,d:abcd=gf(b,d))

for f(b,d) = B(b)D(d) – βδ.

Now by Cauchy-Schwarz we bound this above by

Ea,c,b,d,b′,d′f(b,d)f(b′,d′)

where the expectation is over variables such that abcd = g and ab′cd′ = g. As in the proof that mixing in four steps implies mixing in three, we can rewrite the last two equations as the single equation bcd = b′cd′.

The fact that the same variable c occurs on both sides of the equation is what gives rise to conjugacy classes. Indeed, this equation can be rewritten as

c-1(1∕b)b′c = d(1∕d′).

Performing the following substitutions: b = x,b′ = xh,d′ = y we can rewrite our equation as

d = c-1hcy.

Hence we have reduced our task to that of bounding

Ef(x,C(h)y)f(xh,y)

for uniform x,y,h.

We can further replace y with C(h)-1y, and rewrite the expression as

Ex,y(f(x,y)Ehf(xh,C(h-1)y)).

This is at most

(Ex,yf2(x,y))E x,y,h,h′f(xh,C(h-1)y)f(xh′,C(h′-1)y).

Recalling that f(b,d) = B(b)D(d) – βδ, and that E[f] = βδ, the first factor is at most 1. The second can be rewritten as

Ex,y,h,h′f(x,y)f(xh-1h′,C(h′-1)C(h)y)

replacing x with xh-1 and y with C(h-1)-1y = C(h)y.

Again using the definition of f this equals

Ex,y,h,h′B(x)D(y)B(xh-1h′)D(C(h′-1)C(h)y) – β2δ2.

Now Lemma 6 guarantees that the distribution (x,y,xh-1h′,C(h′-1)C(h)y) is 1∕|G|Ω(1)-close in statistical distance to the uniform distribution over G4, and this concludes the proof. QED

References

[Bar89]    David A. Mix Barrington. Bounded-width polynomial-size branching programs recognize exactly those languages in NC1. J. of Computer and System Sciences, 38(1):150–164, 1989.

[BNP08]    László Babai, Nikolay Nikolov, and László Pyber. Product growth and mixing in finite groups. In ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 248–257, 2008.

[Gow08]    W. T. Gowers. Quasirandom groups. Combinatorics, Probability & Computing, 17(3):363–387, 2008.

[GV15]    W. T. Gowers and Emanuele Viola. The communication complexity of interleaved group products. In ACM Symp. on the Theory of Computing (STOC), 2015.

[KMR66]   Kenneth Krohn, W. D. Maurer, and John Rhodes. Realizing complex Boolean functions with simple groups. Information and Control, 9:190–195, 1966.

Ditch your family and come to FOCS

In a week I will be attending the FOCS conference. As usual, I find the program very interesting, and look forward to the talks. Hopefully not every one of them will be over my head; I hope to write a report about the talks later. It’s also a great fortune that I will be able to get there by train!

Workshops and the end of the celebration for Avi Wigderson’s birthday are on Saturday. The main conference starts on Sunday, and lasts until Tuesday. Monday is Columbus Day, a national holiday. To state the obvious, Saturday, Sunday, and Monday are days when schools, day cares, and other things that allow for work are not available. Couldn’t we hold conferences during week-days, like other regular events such as NSF panels? To be clear this has nothing specific to do with this FOCS, but is a general theme. Which may be part of the reason why certain groups of computer scientists are called minorities.

Sure, holding a conference during week-days means that you have to skip “work.” But isn’t attending the conference at least as important work? And I have never met an attendee who wouldn’t jump for joy if they had a valid excuse to skip a lecture or a committee meeting, nor, but maybe I have been lucky, a dean or department chair who would obstruct attendance.

Exercise, diet, and sleep improve brain skills (and health)

This semester I am teaching 80 undergraduates Theory of Computation. I love the material and so every minute is precious, but I decided to sacrifice a few for a quick illustration of the title. After all, I thought to myself, it *is* my job to know, use, and disseminate teaching techniques that improve the students’ performance. So why shouldn’t I tell them the benefits of cardio exercise on learning? So this morning I scrambled together a few slides which you can see here.

I plan to add much more in future versions, but it is a euphemism to say that I am not an expert in these areas. So I’d very much appreciate any pointers, especially to what are the landmark papers in these areas.

Paper X, on the ArXiv, keeps getting rejected

Paper X, on the ArXiv, keeps getting rejected. Years later, Paper Y comes up and makes progress on X, or does something closely related to X. Y may or may not cite X. Y gets published. Now X cannot get published because the referees do not see what the contribution of X is, given that Y has been published and that in light of Y X is not new.

The solution in my opinion, following a series of earlier posts the last one of which is this, is to move the emphasis away from publication towards ArXiv appearance. Citations should refer to the first available version, often the ArXiv one. Journals/conferences can still exist in this new paradigm: their main important job would be to assign badges to ArXiv papers.

Obviously, this plan does not work for the entities behind current journals/conferences. So they enforce the status quo, and in the most degrading way: by forcing authors to fish out, maintain, and format useless references.

Hokuto No Ken and growing up in Italy

It seems that the Hokuto No Ken videogame that should have been made decades ago is finally being made. Thanks to Marco Genovesi for sending me this link. (More about Marco later on this blog.)

I consider watching the Hokuto No Ken series (excluding the more recent garbage) one of the most significant artistical experiences of my life, something that also makes me understand how can some people be so passionate about Dante or Homer. And, if you grow up in Italy there is a special treat for you. You can watch a version where the words are dubbed, but the soundtrack and the screams are from the original Japanese. By contrast, the English-speaking audience can either watch the Japanese version with subtitles — and I always hate subtitles — or they can watch an English dubbed version. I once happened to get a glimpse of the latter and I was horrified: The masterful soundtrack has been replaced by a very cheap synth, not to mention the screams. Compare this to this.

csrankings


csrankings
provides an objective ranking of CS departments. It is the revamped version of a previous system which I followed also because it did not include NEU. The new one does. But there are reasons slightly more subtle than the fact that NEU ranks 9 in “theory” — or 11 if you remove “logic and verification”, in both cases beating institutions which are way above NEU in other rankings — why I think having this information around is very valuable. Unobjective rankings tend to be sticky and not reflect recent decay or growth. And if one still wants to ignore data at least now one knows exactly what data is being ignored.

One dimension where I would like the system to be more flexible is in the choice of venues to include in the count. For example, I think journals should be added. Among the conferences, CCC should be included, as the leading conference specialized in computational complexity. I think the user should be allowed to select any weighted subset of journals and conferences.

Democracy!

I strongly recommend playing Oiligarchy a brilliant online game about American politics from the point of view of the oil industry. You should play to the end to see which ending you get: There are four, though one of them may be unattainable. But at the very least make sure to get to a presidential election. Its depiction is memorable. (Incidentally, many other games on the website are well worth a look.)

Turning to what is unfortunately not a videogame, this time we have a candidate, let us call them candidate X, whom many think is exceptionally unqualified to be President. I disagree in an unimportant way. I am not sure X really is more unqualified, or more dangerous, than some of the other candidates in recent history, including at least one who actually became President for two terms. The one distinctive feature of X seems to be that X is more colorful and more openly arrogant than other candidates. But I don’t find this to be a substantial difference. I certainly wouldn’t mind it too much if what X said made any sense at all.

I also suspect that X and their entourage know well that the chance of X winning the election is negligible, but want to rake in and maximize publicity, according to the old dictum that there is no bad publicity.

Bounded indistinguishability

Countless papers study the properties of k-wise independent distributions, which are distributions where any k bits are uniform and independent. One property of interest is which computational models are fooled by such distributions, in the sense that they cannot distinguish any such distribution from a uniformly random one. Recently, Bazzi’s breakthrough, discussed earlier on this blog, shows that k = polylog(n) independence fools any polynomial-size DNF on n bits.

Let us change the question. Let us say that instead of one distribution we have two, and we know that any k bits are distributed identically, but not necessarily uniformly. We call such distributions k-wise indistinguishable. (Bounded independence is the special case when one distribution is uniform.) Can a DNF distinguish the two distributions? In fact, what about a single Or gate?

This is the question that we address in a paper with Bogdanov, Ishai, and Williamson. A big thank you goes to my student Chin Ho Lee for connecting researchers who were working on the same problems on different continents. Here at NEU the question was asked to me by my neighbor Daniel Wichs.

The question turns out to be equivalent to threshold/approximate degree, an influential complexity measure that goes back to the works by Minsky and Papert and by Nisan and Szegedy. The equivalence is a good example of the usefulness of duality theory, and is as follows. For any boolean function f on n bits the following two are equivalent:

1. There exist two k-wise indistinguishable distributions that f tells apart with advantage e;

2. No degree-k real polynomial can approximate f to pointwise error at most e/2.

I have always liked this equivalence, but at times I felt slightly worried that could be considered too “simple.” But hey, I hope my co-authors don’t mind if I disclose that it’s been four different conferences, and not one reviewer filed a complaint about that.

From the body of works on approximate degree one readily sees that bounded indistinguishability behaves very differently from bounded independence. For example, one needs k = Ω(√ n) to fool an Or gate, and that is tight. Yes, to spell this out, there exist two distributions which are 0.001 √ n indistinguishable but Or tells them apart with probability 0.999. But obviously even constant independence fools Or.

The biggest gap is achieved by the Majority function: constant independence suffices, by this, while linear indistinguishability is required by Paturi’s lower bound.

In the paper we apply this equivalence in various settings, and here I am just going to mention the design of secret-sharing schemes. Previous schemes like Shamir’s required the computation of things like parity, while the new schemes use different types of functions, for example of constant depth. Here we also rely on the amazing ability of constant-depth circuits to sample distributions, also pointed out earlier on this blog, and apply some expander tricks to trade alphabet size for other parameters.