Hi Ryan, For instance, MAXCUT on bounded degree graphs can be approximated better than the Goemans-Willamson constant … This image is only for illustrative purposes. I The example demonstrates that general, non-sinusoidal signals can be represented as a sum of sinusoids. 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). (video), 10.2 Digression to boosting, experts, dense models, and their quantum counterparts You may have noticed the population parameter that describes the spread of the data, the variance, is squared. 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. But avoid …. I was trying to say that if you wanted to look at an actual proof and guess whether it’s likely to implicitly yield such a property P, the clearest markers of trouble are the use of Chernoff+Union bound arguments. Can you find applications for this conjecture in cryptography? Ryan O’Donnell’s problems above present one challenge to this viewpoint. The sum of squares is one of the most important outputs in regression analysis. Understand the role of noise in the performance of the SOS algorithm. (Suggested by Ryan O’Donnell) Let be the vertex graph on where we connect every two vertices such that their distance (mod ) is at most for some constant . It is basically the addition of squared numbers. (Here the word “probably” is encompassing two things: 1. Some of the topics we covered included the SDP based algorithms for problems such as Max-Cut, Sparsest-Cut, and Small-Set Expansion, lower bounds for Sum-of-Squares: 3XOR/3SAT and planted clique, using SOS for unsupervised learning, how might (and of course also might not) the SOS algorithm be used refute the Unique Games Conjecture, linear programming and semidefinite programming extension complexity. I have suggested that the main reason that a “robust” proof does not translate into an SOS proof is by use of the probabilistic method, but this is by no means a universal law and getting better intuition as to what types of arguments do and don’t translate into low degree SOS proofs is an important research direction. i) with negligible probability, H is miraculously satisfiable; But, that said, this is a necessary condition for not having a proof in general, but it’s not a sufficient condition even for not having SOS proofs. 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 ). We focus on a regression problem with n1 observations and p1 covariates. (video), 2.3. When these component values are squared and summed over all the observations, terms called sums of squares are produced. automatizable ones) that are noise-robust and are stronger than SOS for natural combinatorial optimization problems? It’s because probably no such proof exists. For instance, MAXCUT on bounded degree graphs can be approximated better than the Goemans-Willamson constant via a combination of SDP rounding and local search. SOS, Cryptography, and . [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. Lecture on sums of squares Carl L ondahl March 22, 2011 Question: Which integers n can be represented n = a2 + b2 Lemma 1. 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. Sum of Squares … (video), 5.1. between sum of squares and semidenite programming is the following. 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. Lecture Notes for STATISTICAL METHODS FOR BUSINESS II BMGT 212 Chapters 13 & 14 Professor Ahmadi, Ph.D. Department of Management Revised February 2005 . Can the SOS algorithm give any justification to this intuition? 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? The sum-of-squares algorithm maintains a set of beliefs about which vertices belong to the hidden clique. While sum of squares SDP relaxations yield the best known approximations for CSPs, the same is not known for bounded degree CSPs. But either way, now that we've calculated it, we can actually figure out the total sum of squares. What is the right way to define noise robustness in general? In a regression analysis , the goal … 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 . (pdf version) ii) with (presumably) low (inverse-poly) probability, H is unsatisfiable but for an unusually simple reason (e.g., you happened to pick all 8 possible SAT constraints on the literals associated with x1,x2,x3); Sum of Squares Upper Bounds, Lower Bounds, and Open Questions Boaz Barak December 10, 2014. We deal with the ‘easy’ case wherein the system matrix is full rank. 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. Can you do this with arbitrarily small ? Here “very probably” ONLY refers to the probability that my-instance.cnf has the typical amount of expansion. I just gave the final lecture in my seminar on Sum of Squares Upper Bounds, Lower Bounds, and Open Questions. (video), 5.6. 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.). A natural numberncould be written as a sum of the squares of two integers if and only if every prime factorpofnwhich is of the form 4k+3 enters the canonical decomposition ofnto an even degree. Lower bounds—Unique Games Conjecture, 4.1. The nonlinear maximization problem is 1-dimensional, not 3. 2. The lecture on why we sum squares had to do with the numerator of the variance formula. 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? (video), 1.2. This is one of over 2,200 courses on OCW. 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*. Section 5. iii) with (presumably) all the rest of the probability, H is unsatisfiable but *there is no succinct reason why it’s unsatisfiable*. The set of vertices with least expansion is an arc. (Re my-instance.cnf, let me add that “very probably” there is no low-degree SOS proof that it’s unsatisfiable. Extend this to a quasipolynomial time algorithm to solve the small-set expansion problem (and hence refute the small set expansion hypothesis). 1.1. Can you extend this to larger dimensions? 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. (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.). P (Y^ Y )2: variation explained by the model 4 P At first we thought: “Sure. Give a polynomial-time algorithm that for some sufficiently small , can (approximately) recover a planted -sparse vector inside a random subspace of dimension . Can SOS shed any light on this phenonmenon? 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. (video), 10.1 Is sos an “optimal algorithm”? Can we give any interesting applications of this? doesn’t also show that an instance has a low degree SOS proof of unsatisfiability w.h.p I guess to see if what I say makes sense one should see if one can find a natural way to extract the existence of an Omega(n) degree SOS proof of unsatisfiability from the chernoff+union bound argument. 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. Dictionary learning via tensor decomposition [Update: Prasad notes that the first problem for Max-Cut actually is solved as stated, since the paper also shows that the SDP integrality gap is better than  for Max-Cut on bounded degree graphs. Going back to random 3SAT, when you pick a random instance H, one of three things can happen: Probably not, right? SOS and the unit sphere—Sparse vectors, tensor decomposition, dictionary learning, and quantum separability, 7.2. (video), 7.4. The spread of the data could be described by the range, the maximum … (video), 3.3. Sum of squared deviations about the grand mean across all N observations Sum of squared ... •To make the sum of squares comparable, we divide each one by their associated degrees of freedom •SST G = k - 1 (3 - 1 = 2) • The effective application of linear regression is expanded by data transformations and diagnostics. In particular, is there an SOS proof that the graph constructed by Capalbo, Reingold, Vadhan and Wigderson (STOC 2002) is a “lossless expander” (expansion larger than )? Here local search refers to improving the value of the solution by locally modifying the values. Sum of Squares Sum of squares refers to the sum of the squares of numbers. (Basically, there’s a spectral proof. 1) Regarding your first comment, I am trying to understand if this is a technical or philosophical issue. Can you give a quasipolynomial time algorithm that works when has dimension ? Can we get any formal justifications to this intuition? The sum of squares got its name because it is calculated by finding the sum of the squared differences. 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.