As an intermediate step, settle Khot-Moshkovitz’s question whether for an arbitrarily large constant , the Max-2-LIN instance they construct (where the degree (for some constant ) SOS value is ) has actual value at most . I thought I might post here the list of open problems I ended with- feel free to comment on those or add some of your own below. A major issue in cryptography is (to quote Adi Shamir) the lack of diversity in the “gene pool” of problems that can be used as a basis for public key encryption. Sum-of-Squares seminar: lecture notes and open problems, Sum of Squares Upper Bounds, Lower Bounds, and Open Questions, http://www.cs.cmu.edu/~odonnell/my-instance.cnf, A different type of pseudo | Windows On Theory. Can we use a conjectured optimality of SOS to give public key encryption schemes? i) with negligible probability, H is miraculously satisfiable; Sum of Squares … In what follows we introduce the ordinary least squares (OLS) approach which basically consists in minimizing the sum of squares of the distance between the observed values y iand the predicted values at x iunder the linear model. • The general linear model — an extension of least-squares linear (pdf version) I do believe that part of the “reason” is that when you do the Chernoff+union bound argument then if you open up Chernoff to a bound on moments, then you need to consider very large moments (of linear size) and this is the reason this proof doesn’t translate into an SOS proof, or perhaps more accurately, this is the reason the proof that an instance is unsatisfiable w.h.p. Prove that there is a function that passes the test with for some but such that for every constant and function of the form where a polynomial of degree at most , . The nonlinear maximization problem is 1-dimensional, not 3. In contrast note that we do not believe that this assertion is actually true in general. For every there is a such that if we let be the graph with vertices corresponding to where we connect vertices if their Hamming distance is at most , then for every subsets of satisfying , there is an edge between and . Sum of Squares Sum of squares refers to the sum of the squares of numbers. If you could somehow efficiently certify that a particular 3SAT instance with Cn clauses was both: a) generated at random; b) was not atypical of this random generation; *then* I’d believe that Chernoff-bound-proof-complexity had something to do with SOS-refutability-of-most-random-3SAT instances. Understand the one-way and two-way ANOVA models. And indeed SOS can find it, as can a suitably set up “GW/basic SDP”.) What is the right way to define noise robustness in general? Grothendieck (pdf version) MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.. No enrollment or registration. Lower bounds—Planted Clique Going back to random 3SAT, when you pick a random instance H, one of three things can happen: Max Cut (pdf version), 6.1 Flag algebras and extremal combinatorics. Finding a sparse vector in a subspace Section 5. In a regression analysis , the goal … The notes are very rough, undoubtedly containing plenty of bugs and tpyos, but I hope people would nd them useful Lower bounds—Max Cut (video), 2.1. Indeed, I distinguish between two kinds of SOS lower bounds: a) those, like Grigoriev’s “Knapsack” lower bound and random-3XOR-perfect-satisfiability lower bound, where low-degree SOS fails even though we know simple proofs in other proof systems; b) those, like the random-3SAT lower bounds or the planted clique lower bounds, where low-degree SOS fails because *we expect that every proof system fails*. That is a totally irrelevant issue. Despite learning no new information, as we invest more computation time, the algorithm reduces uncertainty in the beliefs by making them consistent with increasingly powerful proof systems. I actually think that the Gregoriev/Schoenebeck pseudo-distribution do satisfy that the objective value is m with probability 1 (since its expectation is m, and it cannot be larger than m, this must hold if it was an actual distribution, and I think that E (val-m)^2 = 0 can be shown to hold). Sum of Sinusoidal Signals Time-Domain and Frequency-Domain Non-sinusoidal Signals as Sums of Sinusoids I If we allow infinitely many sinusoids in the sum, then the result is a square wave signal. (video), 5.6. So it's going to be equal to 3 minus 4-- the 4 is this 4 right over here-- squared plus 2 minus 4 squared plus 1 minus 4 squared. The broader themes these questions are meant to explore are: Show that for every constant there is some and a quasipolynomial () time algorithm that on input a subspace , can distinguish between the case that contains the characteristic vector of a set of measure at most , and the case that for every . This image is only for illustrative purposes. Objectives Understand analysis of variance as a special case of the linear model. Lecture 7: Hypothesis Testing and ANOVA. Can the SOS algorithm give any justification to this intuition? (pdf version), 8.1. This can be approximated to a 3/4 factor using only local search (no LP/SDP relaxations), and some natural relaxations have a 1/2 integrality gap for it. (Here the word “probably” is encompassing two things: 1. What do you say if I take the instance H and add “OBJECTIVE = k” as an SOS constraint?” I kind of feel that in many cases, SOS will now return “infeasible” for this augmented instance. The problem is that this SOS proof will have degree comparable to the degree of Q which is of course very large. (pdf version) Can you find applications for this conjecture in cryptography? between sum of squares and semidenite programming is the following. (Basically, there’s a spectral proof. The lecture on why we sum squares had to do with the numerator of the variance formula. Incidentally, once m >> n^2, there are even simple *local* reasons for unsatisfiability (whp). Analysis of Variance The analysis of variance is a central part of modern statistical theory for linear models and experimental design. (video), 7.2. SOS and the Unique Games conjecture—a love hate relationship, 9.4. Set z = a+ bi;w = c+ di Here local search refers to improving the value of the solution by locally modifying the values. It is basically the addition of squared numbers. To try to say it another way, imagine an alternate universe where there *is* a low-degree SOS proof of Chernoff bounds or, Chernoff+union-bounds, or whatever. automatizable ones) that are noise-robust and are stronger than SOS for natural combinatorial optimization problems? Despite learning no new information, as we invest more computation time, the algorithm reduces uncertainty in the beliefs by making them consistent with increasingly powerful proof systems. (pdf version) If you think this can’t be done, even showing that the algorithm doesn’t beat would be very interesting. I personally swear to you on my honor that I generated it at random, in the usual way. Here “very probably” ONLY refers to the probability that my-instance.cnf has the typical amount of expansion. Can you extend this to larger dimensions? (see also this previous post). The reduction will not be “complete” in this case, since it will have more than exponential blowup and will not preserve SOS solutions but I still view this as an interesting step. Linear Least-Squares Regression 2 2. But it’s not because you, Boaz Barak, do not know how to prove the Chernoff bound (you do). Let me try to elaborate more, in the hope of not confusing things even further. Re point 2, indeed I think giving a “probability 1” lower bound for random 3AND is a great suggestion for another open problem. Indeed, many of our candidates for public key encryption (though not all— see discussion in [Applebaum,Barak, Wigderson]) fall inside (or ). Therefore, the total variability in the data can be partitioned into a sum of squares of the difierences between the treatment averages and the grand average, plus a sum of squares of the difierences of observations within treatments from the treatment average. Then we can use Markov to argue that the probability for every x that x satisfies f is 100^{-n} and so we can use the union bound over all 2^n choices for x. (pdf version) That is, one defines some non-negative polynomial Q (of degree O(n) ) that takes as input both the coefficients f defining the formula and the n variables x_1…x_n defining the assignments, and you show that on one hand for fixed x and random f, E Q(f,x)=1 and on the other hand if x satisfies f then Q(f,x) > 100^n . SOS and the unit sphere—Sparse vectors, tensor decomposition, dictionary learning, and quantum separability, 7.2. (Suggested by Prasad Raghavendra.) Hi Ryan, Sums of Squares and ANOVA (LECTURE NOTES 13) 255 6.5 Sums of Squares and ANOVA We look at an alternative test, the analysis of variance (ANOVA) test for the slope parameter, H 0: m= 0, of the simple linear model, Y = b+ mX+ ; where, in particular, is N(0;˙2), where the ANOVA table is Source Sum Of Squares Degrees of Freedom Mean Squares You may have noticed the population parameter that describes the spread of the data, the variance, is squared. 1) Regarding your first comment, I am trying to understand if this is a technical or philosophical issue. Extend this to a quasipolynomial time algorithm to solve the small-set expansion problem (and hence refute the small set expansion hypothesis). Understanding generalization requires rethinking deep learning? (pdf version), 5.3. 2 SST= SS Tr+ SSE The notations SS I do agree the latter is often more natural to think of. When these component values are squared and summed over all the observations, terms called sums of squares are produced. SOS, Cryptography, and . A random graph with a hidden clique. (pdf version) Ryan O’Donnell’s problems above present one challenge to this viewpoint. iii) with (presumably) all the rest of the probability, H is unsatisfiable but *there is no succinct reason why it’s unsatisfiable*. Another way to state that a pseudo expectation E satisfies the constraint P=0 with probabilty 1 (when P is a polynomial) is to require that E P^2 = 1. Are there natural stronger proof systems than SOS (even non Just take the Grigoriev/Schoenebeck pseudodistribution; it’s easy to see that when you look at the pseudoexpectation of the objective function, it’ll be exactly (1/4)m.” But… I do not know how to show that this pseudoexpectation will actually SOS-satisfy the constraint “OBJ = (1/4)m”. (pdf version) (Indeed, this may be related to the planted clique question, as these tools were used  to construct the best known Ramsey graphs.). Whenever you prove that a random formula f is unsatisfiable w.h.p., you actually prove that w.h.p. This is not meant to be a complete or definitive list, but could perhaps spark your imagination to think of those or other research problems of your own. The Bayesian view of pseudo distributions. SOS and the unit sphere—Sparse vectors, tensor decomposition, dictionary learning, and quantum separability (pdf version) (video), 5.1. The sum-of-squares algorithm maintains a set of beliefs about which vertices belong to the hidden clique. (video), 10.1 Is sos an “optimal algorithm”? Don't show me this again. Show that there is some constant such that the degree- SOS problem can distinguish between a random graph and a graph in which a clique of size was planted for some , or prove that this cannot be done. For starters, I do believe in the “average case NP\neq coNP” conjecture of Feige et al, so I do not believe there is *any* succinct proof for the instance you generated (even without actually seeing it – I got a 404 error at http://www.cs.cmu.edu/~odonnell/my-instance.cnf ). Least Squares with Examples in Signal Processing1 Ivan Selesnick March 7, 2013 NYU-Poly These notes address (approximate) solutions to linear equations by least squares. On the other hand, I do not know how to show that a random 3AND instance has objective value m/4 “with probability 1”. It sometimes seems as if in the context of combinatorial optimization it holds that “”, or in other words that all proof systems are automatizable. The sum of squares got its name because it is calculated by finding the sum of the squared differences. Indeed, instead of using SOS to maximize an objective subject to (typically) X_i^2 = X_i for all i, you should try all SOS-feasibility instances of the form constraints + “OBJ = k” for all 0 <= k <= m. Seems quite natural, no? This is one of over 2,200 courses on OCW. For instance, MAXCUT on bounded degree graphs can be approximated better than the Goemans-Willamson constant … Lecture 14 Simple Linear Regression Ordinary Least Squares (OLS) Consider the following simple linear regression model Y i = + X i + "i where, for each unit i, Y ... Total Variations (Sum of Squares) P (Y i Y )2: total variation: (n 1) V(Y). (pdf version) That is, we choose as random Gaussian vectors, and the algorithm gets an arbitrary basis for the span of . (pdf version) The small set expansion hypothesis But avoid …. Related to this: is there a sense in which SOS is an optimal noise-robust algorithm or proof system? (You can let Q(x,f) be the twice the fraction of satisfied constraints raised to the power 10n.) The notion of pseudo-distributions gives rise to a computational analog of Bayesian reasoning about the knowledge of a computationally-bounded observer. And so, the best approach we have for such constructions is to use the probabilistic method so we generate an instance I that is only unsatisfiable with high probability, and so we ourselves don’t have a proof it’s unsatisfiable. Like I said, I am not sure we have a technical disagreement. (This of course can be satisfied vacuously but please bear with me.) Another problem to consider is maximum matching in 3-uniform hypergraphs. (video), 9.2. The fact that you can SOS-prove that 99.9% of random 3SAT instances are unsatisfiable in no way helps you prove that *this specific* H is unsatisfiable. Can we prove this with an SOS proof of constant degree? [Update: William Perry showed a degree 4 proof (using the triangle inequality) for the fact that the least expanding sets a power of the cycle. (video), 9.4. On approaches for proving the Unique Games Conjecture, 10.2 Digression to boosting, experts, dense models, and their quantum counterparts, 10.3 Optimality of sos among similar sized sdp’s for csp’s. And then you know the true optimum is actually at most k-1. SOS framework? Show that for every , rounds of SOS give a approximation for this problem, or rule this out via an integrality gap. Total Sum of Squares TheTotal Sum of Squares(TSS) is the total square deviation of the samples from the overall mean. I The example demonstrates that general, non-sinusoidal signals can be represented as a sum of sinusoids. (Indeed two of the papers we covered are “hot off the press”: the work of  Meka-Potechin-WIgderson on the planted clique problem hasn’t yet been posted online, and the work of Lee-Raghavendra-Steurer on semidefinite extension complexity was just posted online two weeks ago.). (video), 1.2. Give a polynomial-time algorithm that for some sufficiently small , can (approximately) recover a planted -sparse vector inside a random subspace of dimension . 1.1. 7.1. They are extremely rough but I hope they would still be useful, as some of this material is not covered (to my knowledge) elsewhere. Can we get any formal justifications to this intuition? If you think this cannot be done then even showing that the (in fact, even ) SOS program does not solve the unique-games problem (or the norms ratio problem as defined above) would be very interesting. LECTURE NOTES #4: Randomized Block, Latin Square, and Factorial Designs Reading Assignment Read MD chs 7 and 8 Read G chs 9, 10, 11 Goals for Lecture Notes #4 Introduce multiple factors to ANOVA (aka factorial designs) Use randomized block and latin square designs as a stepping stone to factorial designs Understanding the concept of interaction 1. (pdf version) Can you do this with arbitrarily small ? In statistical data analysis the total sum of squares (TSS or SST) is a quantity that appears as part of a standard way of presenting results of such analyses. Show that for every constant , there is some such that rounds of SOS yield an approximation for MAXCUT on graphs of maximum degree . Sum-of-squares over the hypercube Lecture Notes for STATISTICAL METHODS FOR BUSINESS II BMGT 212 Chapters 13 & 14 Professor Ahmadi, Ph.D. Department of Management Revised February 2005 . The sum of squares is one of the most important outputs in regression analysis. (video), 3.2. Please be sure to answer the question.Provide details and share your research! Can SDP relaxations simulate local search? 2. SOS and the Unique Games conjecture—a love hate relationship Understand the role of noise in the performance of the SOS algorithm. Now I didn’t verify this but I think that for a fixed random f we will in fact have an SOS proof to bound this polynomial and show that Q(f,x) << 100^n for every x. Perhaps you ask SOS for the optimum value and it returns to you k = (3/4)m. At first you think to yourself, “OK, well, I know the true maximum is at most k; perhaps I can round my SOS solution to get a genuine solution satisfying close to k constraints.” But what I claim you should actually do next is say, “OK SOS, you claim the optimum value is k, huh? (video), 3.3. This is an important trick. (In a recent, yet unpublished, work with Chan and Kothari, we show that small degree SOS programs cannot distinguish between these two cases.). Lecture on sums of squares Carl L ondahl March 22, 2011 Question: Which integers n can be represented n = a2 + b2 Lemma 1. Introduction I Despite its limitations, linear least squares lies at the very heart of applied statistics: • Some data are adequately summarized by linear least-squares regression. (video), 9.1. If quantum computers are built, then essentially the only well-tested candidates are based on a single problem— Regev’s “Learning With Errors” (LWE) assumption (closely related to various problems on integer lattices). The correctness of the general “AvgNP \not \subset co-NP”-ish type belief about random 3SAT held by most complexity experts; you know, roughly the belief that the Alekhnovitch cryptosystem is secure even against coNP attacks.). Sum of Squares is a statistical technique used in regression analysis to determine the dispersion of data points. Perhaps to justify the security of LWE? (video), 2.3. (pdf version) One barrier for the latter could be that breaking LWE and related lattice problems is in fact in or . (video), 7.5. (pdf version) But either way, now that we've calculated it, we can actually figure out the total sum of squares. Arora–Rao–Vazirani Approximation for Sparsest Cut If we believe that the SOS algorithm is optimal (even in some average case setting) for noisy problems, can we get any quantitative predictions to the amount of noise needed for this to hold? (Note that Kelner and Parrilo have a conjectured approach to achieve this.) Arora–Rao–Vazirani Approximation for Sparsest Cut, 5.1. (BTW the degree of the SOS transitions smoothly from Omega(n) degree for O(n)-constraint formulas to O(1) degree for O(n^{1.5})-constraint formulas; it might be interesting to understand whether there is a natural way to get a similar transition from the chernoff+union bound proof to the spectral proof.). Lower bounds—Unique Games Conjecture, 4.1. (video), 2.2. Whereas once m >> n^{1.5}, your Coja-Oghlans and your Feige-Ofeks have indeed shown that for a random 3SAT instance there is a *comprehensible/succinct* — albeit still global — reason why it’s unsatisfiable. The *flip* side to this is that if you are proving SOS lower bounds, and you show that there is a pseudo-solution where the objective function has pseudo-expectation at least k, even though the true optimum is k' = (1/4 – o(1))m, or even = (1/4)m? Enter your email address to follow this blog and receive notifications of new posts by email. (video), 3.1. In order to certify that some graph G and some value c satisfy max fGc it is enough to exhibit a single bipartition x 2f0,1gnof G such that fG(x) c. 1. σ2=∑i=1n(yi−y¯)2n After determining the center of mass of a data set, the mean, statisticians wanted to have a convenient way to describe the spread of the data. Some of the variation in Y is ‘explained’ by the model and some of it left … Can we prove this with an SOS proof of constant (independent of ) degree? Can you give a quasipolynomial time algorithm for certifying the Restricted Isometry Property (RIP) of a random matrix? Are there SOS proofs for the pseudorandom properties of the condensers we construct in the work with Impagliazzo and Wigderson (FOCS 2004, SICOMP 2006) or other constructions using additive combinatorics? Prove that if this is modified to a single -sized cloud then the reduction would be “sound” in the sense that there would be no integral solution of value larger than . The probability that my-instance.cnf really is unsatisfiable, and not for a flukishly simple reason. The algorithm seems to be inherently noise robust, and it also seems that this is related to both its power and its weakness– as is demonstrated by cases such as solving linear equations where it cannot get close to the performance of the Gaussian elimination algorithm, but the latter is also extremely sensitive to noise. Finding a sparse vector in a subspace, 7.4. Measure of Total Variation • The measure of total variation is denoted by • SSTO stands for total sum of squares • If all Yi’s are the same, SSTO = 0 • The greater the variation of the Yi’s the greater SSTO SSTO= (Yi−Y¯)2 Frank Wood, fwood@stat.columbia.edu Linear Regression Models Lecture 6, Slide 6 Chapter 13 Formulas ... Total Sum of Squares… (video), 10.3 Optimality of sos among similar sized sdp’s for csp’s Re point 1, let me try to clarify things. Cheeger (pdf version) The general rule is that a smaller sum of squares indicates a better model, as there is less variation in the data. 6.1 Flag algebras and extremal combinatorics. Another approach is to try to use techniques from derandomization such as use of additive combinatorics or the Zig-Zag product to obtain “hard to SOS” proofs. Are there natural noise-robust algorithms for combinatorial optimizations that are not captured by the The following statement, if true, demonstrates one of the challenges in proving the soundness of KM construction: Recall that the KM boundary test takes a function and checks if where and have standard Gaussian coordinates that are each correlated for some . We focus on a regression problem with n1 observations and p1 covariates. Say you are an algorithmicist trying to use constant-degree SOS on your favorite CSP-maximization instance H with m constraints. I digress. PS: OK, I didn’t actually make my-instance.cnf , That’s a shame, I was going to assign it as a take home exam . We deal with the ‘easy’ case wherein the system matrix is full rank. Are there natural stronger proof systems that are still automatizable (maybe corresponding to other convex programs such as hyperbolic programming, or maybe using a completely different paradigm)? (video), 7.4. Theorem 2.1: ( [20], [2]) A multivariate polynomial p(x ) in n variables and of degree 2d is a sum of squares if and only if there exists a positive semidenite matrix Q (often called the Gram matrix) such that p(x ) = zT Qz; (4) where z is the vector of monomials of degree up to d I was merely proposing an empirical heuristic as to when we should expect looking at a proof in a paper that, say, a formula from some distribution is unsatisfiable, that this proof would actually show that the SOS algorithm certifies this with low degree, and when we should be worried that this would not be the case. It’s because probably no such proof exists. PS: re point number 2, yes indeed the Grigoriev/Schoenebeck pseudodistribution has objective value m “with probability 1” as you say, which is why I count these as legitimate SOS lower bounds. My point was different. (pdf version) On approaches for proving the Unique Games Conjecture Now, in some cases, such as in the spectral proof that an m^{1.5} clause formula is unsatisfiable, it turns out that the proof establishes a strong property P that also implies the existence of a constant degree SOS proof that f is unsatisfiable. But your last paragraph I don’t understand at all. Dictionary learning via tensor decomposition In particular, if you used >> n^{1.5} random clauses then there will be an SOS proof of unsatisfiability. squares methods, basic topics in applied linear algebra. In case (iii), neither constant-degree SOS, nor poly-size Cutting Planes, nor poly-size Extended Frege, nor even poly-size ZFC (presumably) can show that H is unsatisfiable. If you go to “my homepage / my-instance.cnf” you’ll find a 3SAT instance with n = 10,000 and m = 160,000 I just created. 13-1 Lecture 13 Extra Sums of Squares STAT 512 Spring 2011 Background Reading KNNL: 7.1-7.4 The sum-of-squares algorithm maintains a set of beliefs about which vertices belong to the hidden clique. 2. Sum-of-squares certificates for Max Cut A simple way to describe the power of sum-of-squares for Max Cut is in terms of certificates. Least Squares Procedure The Least-squares procedure obtains estimates of the linear equation coefficients β 0 and β 1, in the model by minimizing the sum of the squared residuals or errors (e i) This results in a procedure stated as Choose β 0 and β 1 so that the quantity is minimized. 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Algorithm give any justification to this intuition about the knowledge of a random formula f is unsatisfiable computational of!