I am on the FOCS program committee. The deadline for submission is in four days, and I look forward to reading in detail about the exciting progress that theory is seeing! (To pick a random exciting paper: this.) This time around I will also run a small private experiment and report on it after the decisions. I was looking briefly for an online system that would allow me to commit to the experiment with the usual cryptographic guarantees, but couldn’t find it. I thought maybe I could commit now to using a definite future value, say tomorrow’s closing value of the S&P500, as a seed for a commitment scheme. It would be overkill in this case, because the experiment I have in mind is not so important, perhaps, but in other scenarios it may be useful to be able to do this without much effort.

# The Day Care Slavery

There will be a time in my life where I won’t have to worry about day cares anymore. That time will be enormously sad, but I will still find a way to rejoice, because it will be the end of the Day Care Slavery. You hear about the surging cost of college, you hear about the gender gap. But you don’t hear as much about the Day Care Slavery.

If you are planning to start a family my best advice is to start looking at day cares before even trying, and calling your top picks the moment you find out you are expecting, before you call anyone else. A typical wait time for a spot in a day care is… two years! (This reminds me of a great scene in an Italian comedy film — Fantozzi, probably not well known outside of Italy — where Fantozzi bribes a doctor to get a spot in the hospital for his daugther’s abortion. The doctor agrees and says it will take two years for the spot. At which Fantozzi replies that gestation lasts only nine months. “But then your case is desperate!” says the doctor.)

Think of a family which is relocating. The hassle of moving pales compared to these wait lists.

Sure, there are other options with a faster turnaround. They are generally incredibly worse. There is also pretty substantive research that quality day cares has a disproportionate effect on life-long “success”. (Success is measured according to typical indicators, whence the quotes.) A quick search pointed me to many interesting papers that back this up. The keyword to look for is “early childhood education”. I’ll just give a few meta-references (that is, popular-press articles that summarize the research findings):

Long-Term Effects of Early Childhood Programs on Cognitive and School Outcomes, Preschool education boosts children’s academic success, research finds, Preschool Education and Its Lasting Effects: Research and Policy Implications, and finally The Sooner The Better: Early Childhood Education, A Key To Life-Long Success.

The day cares must know all of this, because a full-time infant spot in a day care (good Boston-area location) costs more than… $3K/month! Higher than any rent I have ever paid (and I spent one year in a not bad place in Chelsea, Manhattan).

It is not uncommon that one year of day care will cost you more than a year of college. Plus there are many ways in which you can save for college. It is hard but not impossible to get a scholarship for attending a school, but go get one for a preschool. And you can always work to pay for college. Infants can’t do that. College themselves offer many options, like being a teaching or research assistant, to save for education.

What do you get for the hefty tuition of a day care, by the way? You get to shut the infant in a windowless basement for the entire day. This is the usual issue that they are liable if a child trips outside and scratches a knee, but no method was yet FDA approved to measure the brain damage ensuing from spending 10 hours in a basement during summertime. (Naturally, they promise that the child will go out multiple times *every* day.) You also have to bring lunch, and provide diapers and wipes.

Frankly, I find all of this ridiculous. Think about it. When your child hits 5 years, they are guaranteed a free spot in public schools. Until then, you’re on your own. You are not guaranteed a spot in a day care, and if you are lucky to get it it costs a fortune. The government’s support in this crucial juncture is this: you can save taxes on $5K spent towards day cares. That’s about $1500 bucks, less than half of one month of tuition. And naturally for this you have to fill some extra bureaucracy, and have a little less cash for a year (you get it all back after that). Oh and by the way, if you end up not using the money, sorry, everything’s gone.

Providing quality, affordable day care is one of the most effective ways to achieve many desired goals, including life-long “success” and reducing the gender gap.

# Stoner

The beginning of spring is a busy time in academia. Many committees reach their climax and require immediate action. At the same time, here in Massachusetts the snowstorms that survive climate change carefully destroy schedules that took and will take more weeks to prepare. But one thing that I have managed to do between filling Doodle forms is finish *Stoner*. The book was written and more or less unnoticed in the 60’s; it was then reprinted in the current millennium and became a success. (I owe my own discovery to my mom who shipped me the Italian translation as a present.)

If you ever wonder about the meaning of life, especially academic life, I recommend this book. I almost put it away because I found the beginning a little boring, but I am happy I did not, even though the book isn’t exactly uplifting.

# Black Viper

Over at the Amigospodcast, dreamkatcha posted a series of five posts on the defunct Italian software house Lightshock software. This post is about Black Viper, a game which I coded up back in the 90’s. The last post is about what happened to us afterwards. I have contributed to the posts the following (which you can also read there among nice pictures):

Black Viper is the second game I coded up, after Nathan Never. I vividly remember when I turned down an offer to work on Nathan Never 2. In hindsight, I think it’s been a mistake. True, my experience with Nathan Never had not been very pleasant. The producers did not pay half of what was promised on the contract (something for which I subsequently sued them, in vain). The person who most closely directed the works did not have much experience with games, but understandably was mostly interested in the publicity that would come from the title (Nathan Never is a popular comic book in Italy). I was also threatened extortion if I didn’t finish the game in time.

But on the positive side, after some difficulty they had shipped me an A3000T to complete the game. And most importantly of all, the game got down in six months! Black Viper instead took three ominous years.

But when I made the decision I was 15 and I really had no idea. I had always wanted to work on a beat’em up, and at some point Marco Genovesi and I even had a small demo. My impression is that we ended up working on a bike game because neither of us liked it. We did like the post-apocalyptic atmosphere though.

So we started working on this project again during our high school (the game finally hit the shelves during my first year of college). Our initial name was “Dark Blade” and the game had disproportionate depth, including bets, tournaments, and race against radioactive magma. It was again great to work with Marco, one of the most talented persons I have ever met. We were in the same classroom in high school every day (Italian high school has or at least had a rigid system thanks to which you spent all your time with the same people). Classrooms had desks for two people, and I think the first year Marco and I even shared the desk. If I remember correctly we were both somewhat timid and we naturally ended up together (but here I may be stretching my memory). We would exchange floppy disks in class and generally chat about the game and sketch ideas. Interaction wasn’t always easy, but it was always rewarding.

Our professors took note of what we were doing, and said nothing. I believe in almost any other environment they would have jumped on the students and push them hard to their full potential.

Soon, completing Dark Blade without a however horrible software house proved quite difficult. I don’t remember exactly how we got to Lightshock Software. I think it was through some personal connection. In any case, at some pint this software house materialized, they liked the game, and had some connection to sell it in the broader European market through NEO. So we decided to go with them. We had quite a few fun trips from Rome (where we were based) to Prato (where Lightshock Software was) to discuss the game and our subsequent expansion and world domination. It was also fun to connect with more game developers. They worked with us to make the game better, and to add some extras for the AGA and CD32 versions.

One fond memory I have is when I needed help to hunt down a bug. Basically, the game would occasionally hang up. I had never given that much thought, since it was quite rare. But, of course, the last version of the game systematically crashed exactly at the last level. It’s hard to imagine anything worse. First, reproducing the bug wasn’t the easiest thing. Second, the code was three years old, and it consisted of a single, colossal file in assembly, commented as thoroughly as only a 15-year-old can.

So it was arranged for me to go spend some days at their headquarters and work with Marco Biondi, another Amiga coder. He hosted me at his quaint house in Florence, and together we had a few days of intense debugging. He had no idea of the code whatsoever of course, and was horrified to see certain parts of it which reflected my complete lack of training (my only source had been the Amiga Hardware reference manual, brought to me years ago by Marco Genovesi when I had sprained my ankle jumping from the top of a swing). But Marco Biondi was quite helpful. Eventually, we were able to freeze the machine right before the crash, and we went through one assembly instruction at the time. Amazingly, at some point the instructions became complete gibberish. We exulted. Someone entered the room and proposed something, perhaps going out. I remember Marco Biondi saying no, now “lo teniamo per le palle” (we grab it by the balls).

We looked at how long the gibberish code was, and it turned out to be exactly the length of one of the rectangular elements that made the road. This quickly led us to investigate the code that builds the road. And there I found an assembly subroutine where the register d0 was used instead of a0, which would cause problems if I remember correctly with the carry-around when you increment it. We fixed it and the game didn’t crash anymore. Amazingly, that subroutine was one of the very first which I had written. Throughout the day, we had been playing continuously the same clip of about ten seconds of a heavy metal song.

I think the my share of the sales amounted to the equivalent of $500 today, much less than what I had made with Nathan Never.

Towards the end of Black Viper, and after it, I also worked with other people on a 3D engine. We used things like binary-space-partition trees, and had a demo working both on PC and Amiga. But then it quickly became clear that the only way to even hope to produce anything competitive would be to work on the project full time, which was also difficult because we worked on different parts of Rome and there was no internet. It had been much more convenient to share floppy disks in class with Marco Genovesi!

At some point I also got an offer from Lightshock software to move to their Belluno headquarters to work full time on programming games. They promised a hefty salary and considerable freedom. I thought about it, but in the end I did not go for it. I was in the middle of my college studies which were going well and I didn’t feel like abandoning them (my parents also advised me against). Again I remember the phone call during which I turned them down, perhaps another mistake. I had always wondered what had happened to Lightshock software!

Afterwards I was in touch with other wanna-be software houses, but it was clear that they did not have the capacity that Lightshock software had, and I think they did not end up producing any game.

So I completed my studies, I got interested in mathematics and theoretical computer science, and ended up doing a Ph.D. at Harvard University, and have stayed in the US ever since. I still program, and occasionally I toy with the idea of being involved again with a computer game (besides as a player of course, that has never stopped).

# 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 n^{1+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 ≤ 2^{d} non-zero entries per row (and so a total of ≤ n2^{d} 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 2^{d} 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 n^{ϵ} – 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 n^{2}∕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(n^{2}∕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 _{2}t + 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∕2^{b}) 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(2^{d-b} + 2^{d-2b} + …) = O(2^{d-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∕2^{b}) fraction of the n log ^{*}n edges.

Writing 2^{b} = 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 n^{1+ϵ} non-zero entries, and B has rank n^{1-ϵ′}, for any ϵ and an ϵ′ which goes to 0 when ϵ does.

Their upper bound works as follows. Let h = log _{2}n. Start with the univariate, real polynomial p(z_{1},z_{2},…,z_{h}) 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 z_{i} with x_{i}y_{i} 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(ϵ^{2}). 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 f_{o} at the output gate and fan-in f_{i} at each input gate, then the correspondence in parameters is

rank | = log f_{o} |
||

sparsity | = 2^{fi }. |

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 2^{e} and input fan-in 2^{d}.

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

The available bounds are of the form log f_{o} = n∕f_{i}. For example, for input fan-in f_{i} = n^{α} we have lower bounds exponential in n^{1-α} but not more. Again it can be shown that breaking this tradeoff in certain regimes (namely, log _{2}f_{o} = 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 F_{q} we have the following equalities of distributions:

C(C(a)C(b)) = x^{-1}(y^{-1}ayz^{-1}bz)x = x^{-1}(y^{-1}ayxx^{-1}z^{-1}bz)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 q^{2} (the whole group has size q^{3} – 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 q^{2} 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) → F_{q} is essentially a bijection between conjugacy classes and the field F_{q}. To see this recall that the trace map satisfies the cyclic property:

Tr xyz = Tr yzx.

This implies that

Tr u^{-1}au = 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 F_{q} 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

u_{1} u_{2}

u_{3} u_{4}

you can now with some patience work out the expression

Tr au^{-1}bu = vu_{ 3}u_{4} – u_{3}^{2} + u_{ 4}^{2} – vu_{ 1}u_{2} + u_{1}^{2} – vwu_{ 2}u_{3} + wu_{1}u_{3} – u_{2}^{2} – wu_{ 2}u_{4}.

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, u_{1}u_{4} – u_{2}u_{3} = 1.

Maybe there is some machinery that immediately does that. Lacking the machinery, you can use the equation u_{1}u_{4} – u_{2}u_{3} = 1 to remove u_{4} by dividing by u_{1} (the cases where u_{1} = 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 F_{q}. The probability is 1∕q plus lower-order terms, and then by summing over all a in F_{q} 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

# 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 [Gow08, BNP08]] 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,

|E_{abc=g}A(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

E_{abc=g}A(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,

E_{abcd=g}A(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

|E_{abc=g}A(a)B(b)C(c) – αβγ| = |E_{abc=g}f(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:

(E_{abc=g}f(a)B(b)C(c))^{2} |
≤ (E_{ c}C(c)^{2})(E_{ c}(E_{ab=gc-1}f(a)B(b))^{2}) |
||

= γE_{c}E_{ab=a′b′=gc-1}f(a)B(b)f(a′)B(b′) |
|||

= γE_{ab=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 -α^{2}β^{2}. 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′) = 1_{G}. 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^{-1}gh : 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

E_{a,c}(A(a)C(c)E_{b,d:abcd=g}f(b,d))

for f(b,d) = B(b)D(d) – βδ.

Now by Cauchy-Schwarz we bound this above by

E_{a,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^{-1}hcy.

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)^{-1}y, and rewrite the expression as

E_{x,y}(f(x,y)E_{h}f(xh,C(h^{-1})y)).

This is at most

(E_{x,y}f^{2}(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

E_{x,y,h,h′}f(x,y)f(xh^{-1}h′,C(h′^{-1})C(h)y)

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

Again using the definition of f this equals

E_{x,y,h,h′}B(x)D(y)B(xh^{-1}h′)D(C(h′^{-1})C(h)y) – β^{2}δ^{2}.

Now Lemma 6 guarantees that the distribution (x,y,xh^{-1}h′,C(h′^{-1})C(h)y) is 1∕|G|^{Ω(1)}-close in statistical distance to the uniform distribution over G^{4}, and this concludes the proof. QED

### References

[Bar89] David A. Mix Barrington. Bounded-width polynomial-size branching programs recognize exactly those languages in NC^{1}. 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.