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. Sum of Squares Sum of squares refers to the sum of the squares of 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. 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). (video), 2.3. (video), 2.1. Can we understand better the role of the feasible interpolation property in this context? The probability that my-instance.cnf really is unsatisfiable, and not for a flukishly simple reason. 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. (pdf version) We used the same device to estimate models with autocorrelation the ˆ di⁄erenced model … In a regression analysis , the goal … For example, can we show (without relying on the UGC) that for every predicate , and , if there is an -variable instance of MAX- whose value is at most but on which the degree SOS program outputs at least , then distinguishing between the case that a CSP- instance as value at least and the case that it has value at most is NP-hard? A random graph with a hidden clique. (pdf version) Re the number of constraints, yes, I’m assuming we’re working with m = Cn constraints for C a large constant. Lecture 14alysis of Variance : An I. You may have noticed the population parameter that describes the spread of the data, the variance, is squared. The total sum of squares is the sum of squared ts plus the sum of squared residuals. (video), 5.2. But surely that certification task is not doable (nor even well-posed, really). (video), 7.1. (video). If you think the above cannot be done, even showing that the degree (or even better, ) SOS program cannot achieve this, even for the more general Max-2-LIN problem, would be quite interesting. The problem is that this SOS proof will have degree comparable to the degree of Q which is of course very large. (pdf version) The squared terms could be 2 terms, 3 terms, or ‘n’ number of terms, first n even terms or odd terms, set of natural numbers or consecutive numbers, etc. If m = a2 + b2;n = c2 + d2 =)mn = x2 + y2 Proof. 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. 2. 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? Arora–Rao–Vazirani Approximation for Sparsest Cut, 5.1. 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. Can you extend this to larger dimensions? (pdf version) 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. 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. We deal with the ‘easy’ case wherein the system matrix is full rank. The notes are very rough, undoubtedly containing plenty of bugs and tpyos, but I hope people would nd them useful Welcome! (video), 3.3. 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 . 1) Regarding your first comment, I am trying to understand if this is a technical or philosophical issue. Some intermediate steps that could be significantly easier are: the Khot-Moshkovitz construction is a reduction from a -CSP on variables that first considers all -sized subsets of the original variables and then applies a certain encoding to each one of those “cloud”. 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. I think this is a great point (which I keep forgetting and should have emphasized in the seminar more) that if you want to come up with an unsatisfiable SAT instance I for which there exists no short proof that I is unsatisfiable, then *you* better have no proof for this as well. The Bayesian view of pseudo distributions. I do agree the latter is often more natural to think of. 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 . Can we give any interesting applications of this? Thanks to everyone that participated in the course and in particular to the students who scribed the notes (both in this course and my previous summer course) and to Jon Kelner and Ankur Moitra that gave guest lectures! Another problem to consider is maximum matching in 3-uniform hypergraphs. (pdf version) Sum-of-squares over general domains Extend this to a quasipolynomial time algorithm to solve the small-set expansion problem (and hence refute the small set expansion hypothesis). On the other hand, I do not know how to show that a random 3AND instance has objective value m/4 “with probability 1”. 7.1. (You can let Q(x,f) be the twice the fraction of satisfied constraints raised to the power 10n.) 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 (pdf version) Deep Double Descent (cross-posted on OpenAI blog), Making TCS more connected / less insular, Announcing the WiML-T Mentorship Program (guest post), On Galileo Galilei and “denialism” from elections to climate to COVID, Announcing the WiML-T Mentorship Program (guest post), Updated Research Masters programs database by Aviad Rubinstein and Matt Weinberg, MoPS and Junior-Senior Meeting (DISC 2020), Harvard Quantum Initiative postdocoral fellowship at Harvard (apply by December 4, 2020), TCS Visioning Workshop — Call for Participation, Can we understand in what cases do SOS programs of intermediate degree (larger than, Can we give more evidence to, or perhaps refute, the intuition that the SOS algorithm is, Can we understand the performance of SOS in. 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. What is the right way to define noise robustness in general? 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. SOS framework? Preface These are lecture notes for a seminar series I gave at MIT in the fall of 2014. 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. It is defined as being the sum, over all observations, of the squared differences of each observation from the overall mean. Total Sum of Squares TheTotal Sum of Squares(TSS) is the total square deviation of the samples from the overall mean. It is basically the addition of squared numbers. 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. (pdf version) That is, we choose as random Gaussian vectors, and the algorithm gets an arbitrary basis for the span of . (This should be significantly easier to prove than the soundness of the Khot-Moshkovitz construction since it completely does away with their consistency test; still to my knowledge it is not proven in their paper. Can you give a quasipolynomial time algorithm for certifying the Restricted Isometry Property (RIP) of a random matrix? Can we prove this with an SOS proof of constant (independent of ) degree? The lecture on why we sum squares had to do with the numerator of the variance formula. Linear Least-Squares Regression 2 2. Some concrete questions along these lines are: Find some evidence to the conjecture of  Barak-Kindler-Steurer  (or other similar conjectures) that the SOS algorithm might be optimal even in an average case setting. Finding a sparse vector in a subspace, 7.4. (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.). More generally, can we obtain a “UGC free Raghavendra Theorem”? There is an additive relation which states that the total sum of squares equals the sum of the treatment and error sum of squares. Perhaps in economics? Can you do this with arbitrarily small ? 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. ANOVA Table Source of Sum of Degrees of Mean Variation Squares Freedom Square F iii) with (presumably) all the rest of the probability, H is unsatisfiable but *there is no succinct reason why it’s unsatisfiable*. Initially the beliefs have maximum uncertainty and correspond to the uniform distribution but they eventually converge on the correct hidden clique (red edges). I feel like I say this comment every time I chat with Boaz about SOS, but I’ll say it one more time The fact that sublinear-degree SOS cannot refute random 3SAT has **nothing** to do with whether or not “Chernoff+union bound” is SOS-able or not SOS-able. automatizable ones) that are noise-robust and are stronger than SOS for natural combinatorial optimization problems? Sum of Squares is a statistical technique used in regression analysis to determine the dispersion of data points. Lecture 7: Hypothesis Testing and ANOVA. SOS and the unit sphere—Sparse vectors, tensor decomposition, dictionary learning, and quantum separability Can we get any formal justifications to this intuition? Show that there is some and such that for sufficiently small , the degree SOS program for Max-Cut can distinguish, given a graph , between the case that has a cut of value and the case that has a cut of value . 2) Your second point is another excellent one. (video), 9.4. The small set expansion hypothesis By virtue of Grigoriev’s paper, one doesn’t need to have any other “probably-true-but-unsubstantiated” beliefs about hardness-on-average of 3SAT.). 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. This is one of over 2,200 courses on OCW. P (Y^ Y )2: variation explained by the model 4 P While sum of squares SDP relaxations yield the best known approximations for CSPs, the same is not known for bounded degree CSPs. The sum-of-squares algorithm maintains a set of beliefs about which vertices belong to the hidden clique. Can the SOS algorithm give any justification to this intuition? Let me put it another way. (video), 5.6. 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. At first we thought: “Sure. 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) (video), 4.1. 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 . Introduction 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. Re point 2, indeed I think giving a “probability 1” lower bound for random 3AND is a great suggestion for another open problem. On approaches for proving the Unique Games Conjecture One barrier for the latter could be that breaking LWE and related lattice problems is in fact in or . Let me try to elaborate more, in the hope of not confusing things even further. Can you find applications for this conjecture in cryptography? SOS and the Unique Games conjecture—a love hate relationship 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. (pdf version) Can we prove this with an SOS proof of constant degree? 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. Re point 1, let me try to clarify things. (video), 9.2. Enter your email address to follow this blog and receive notifications of new posts by email. 2 SST= SS Tr+ SSE The notations SS But your last paragraph I don’t understand at all. (Re my-instance.cnf, let me add that “very probably” there is no low-degree SOS proof that it’s unsatisfiable. between sum of squares and semidenite programming is the following. The "Sum of Squares" algorithm was independently discovered by researchers from several communities, including Shor, Nesterov, Parrilo, and Lasserre, and is an algorithmic extension of classical mmath results revolving around Hilbert's 17th question. It’s because probably no such proof exists. 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); I must confess I still have no idea what you’re saying with respect to point 1. (video), 2.2. (see also this previous post). But avoid …. Now, I'll do these guys over here in purple. [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. Here “very probably” ONLY refers to the probability that my-instance.cnf has the typical amount of expansion. sum of squares function S can be concentrated, that is written as a function of one parameter alone. i) with negligible probability, H is miraculously satisfiable; 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. Are there natural noise-robust algorithms for combinatorial optimizations that are not captured by the 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. Taking the viewpoint of pseudo-distributions, (which as usual I like to pretend are actual distribubtions) I think you are distinguishing between the case that the objective value is (3/4)m in expectation and the case that it satisfies this with probability 1. 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. 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 and the Unique Games conjecture—a love hate relationship, 9.4. 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. I would suspect the answer might be “no”. 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. (Suggested by Prasad Raghavendra.) The general rule is that a smaller sum of squares indicates a better model, as there is less variation in the data. Related to this: is there a sense in which SOS is an optimal noise-robust algorithm or proof system? Please be sure to answer the question.Provide details and share your research! (pdf version) f has some deterministic property P which implies that f is unsatisfiable. We also have BMS = BSS K−1, EMS = ESS N−K. (video), 3.2. Can SDP relaxations simulate local search? And then you know the true optimum is actually at most k-1. Dictionary learning via tensor decomposition 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*. (video), 7.4. 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. This comment is kind of a joint observation/problem with Sangxia Huang (also based on ideas told to me by Yuan Zhou); hope he doesn’t mind me posting it. The sum-of-squares algorithm maintains a set of beliefs about which vertices belong to the hidden clique. (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 . Extend this to a quasipolynomial time algorithm to solve the unique-games problem (and hence refute the unique games conjecture). The Bayesian view of pseudo distributions 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)? Incidentally, once m >> n^2, there are even simple *local* reasons for unsatisfiability (whp). 1. TSS= X i,j (X ij −X¯)2 = (N−1)s2, where sis the sample SD of all observations together. (pdf version) I personally swear to you on my honor that I generated it at random, in the usual way. Max Cut Say you are an algorithmicist trying to use constant-degree SOS on your favorite CSP-maximization instance H with m constraints. 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. I just gave the final lecture in my seminar on Sum of Squares Upper Bounds, Lower Bounds, and Open Questions. (Basically, there’s a spectral proof. Cheeger I digress. Dictionary learning via tensor decomposition, 8.1. If the system matrix is rank de cient, then other methods are 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. (Here the word “probably” is encompassing two things: 1. When these component values are squared and summed over all the observations, terms called sums of squares are produced. Sum-of-squares over the hypercube Going back to random 3SAT, when you pick a random instance H, one of three things can happen: Ryan O’Donnell’s problems above present one challenge to this viewpoint. Can we use a conjectured optimality of SOS to give public key encryption schemes? • The effective application of linear regression is expanded by data transformations and diagnostics. 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. Or cryptography? Don't show me this again. (This of course can be satisfied vacuously but please bear with me.) Lecture 11: Nested and Split-Plot Designs Montgomery, Chapter 14 1 Lecture 11 – Page 1. (pdf version), 5.3. That is a totally irrelevant issue. Section 5. Asking for help, clarification, or responding to other answers. And indeed SOS can find it, as can a suitably set up “GW/basic SDP”.) 3.3. 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? (pdf version), 6.1 Flag algebras and extremal 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 . Tensor decomposition via sum of squares, 7.5. • The general linear model — an extension of least-squares linear Are there natural stronger proof systems than SOS (even non (video), 7.2. Our goal is to give the beginning student, with little or no prior exposure to linear algebra, a good ground-ing in the basic ideas, as well as an appreciation for how they are used in many SOS, Cryptography, and . But it’s not because you, Boaz Barak, do not know how to prove the Chernoff bound (you do). Understand the one-way and two-way ANOVA models. Grothendieck Thanks for contributing an answer to Cross Validated! (video), 10.1 Is sos an “optimal algorithm”? 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, many of our candidates for public key encryption (though not all— see discussion in [Applebaum,Barak, Wigderson]) fall inside (or ). Find more problems in the area of unsupervised learning where one can obtain an efficient algorithm by giving a proof of identifiability using low degree SOS. This image is only for illustrative purposes. Some of the variation in Y is ‘explained’ by the model and some of it left … Lower bounds—Planted Clique MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.. No enrollment or registration. 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. 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. 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? 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 . In particular, if you used >> n^{1.5} random clauses then there will be an SOS proof of unsatisfiability. Can SOS shed any light on this phenonmenon? Lower bounds—Unique Games Conjecture, 4.1. 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 )? It is the second moment of the data, as the skewness is the third moment. I The example demonstrates that general, non-sinusoidal signals can be represented as a sum of sinusoids. Can you give a quasipolynomial time algorithm that works when has dimension ? Give a polynomial-time algorithm that for some sufficiently small , can (approximately) recover a planted -sparse vector inside a random subspace of dimension . (In a recent, yet unpublished, work with Chan and Kothari, we show that small degree SOS programs cannot distinguish between these two cases.). (pdf version), 8.1. (Indeed, this may be related to the planted clique question, as these tools were used  to construct the best known Ramsey graphs.). Show that the SOS algorithm is optimal in some sense for “pseudo-random” constraint satisfaction problems, by showing that for every predicate , and pairwise independent distribution over , it is NP hard to distinguish, given an instance of MAX- (i.e., a set of constraints each of which corresponds to applying to literals of some Boolean variables ), between the case that one can satisfy fraction of the constraints, and the case that one can satisfy at most fraction of them. (video), 7.5. 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. Tensor decomposition via sum of squares (pdf version) 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. Show that there are some constant and , such that the degree -SOS program yields an approximation to the Sparsest Cut problem. The sum of squares is one of the most important outputs in regression analysis. 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? Sum of Squares … For instance, MAXCUT on bounded degree graphs can be approximated better than the Goemans-Willamson constant … (video), 10.3 Optimality of sos among similar sized sdp’s for csp’s 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). It sometimes seems as if in the context of combinatorial optimization it holds that “”, or in other words that all proof systems are automatizable. Finding a sparse vector in a subspace The notion of pseudo-distributions gives rise to a computational analog of Bayesian reasoning about the knowledge of a computationally-bounded observer. 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. Analysis of Variance The analysis of variance is a central part of modern statistical theory for linear models and experimental design. 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. (pdf version) Lecture on sums of squares Carl L ondahl March 22, 2011 Question: Which integers n can be represented n = a2 + b2 Lemma 1. (video), 9.1. II. (pdf version) (pdf version) 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. 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 point is that in this regime, a random 3SAT instance is not just unsatisfiable with high probability — rather, experts like Pudlak or Santhanam or Feige or whoever would conjecture that “with high probability it’s unsatisfiable for an *incomprehensible/non-succinct global reason*”. Set z = a+ bi;w = c+ di (pdf version) (pdf version) So let's do that. Chapter 13 Formulas ... Total Sum of Squares… (pdf version) squares methods, basic topics in applied linear algebra. They are extremely rough but I hope they would still be useful, as some of this material is not covered (to my knowledge) elsewhere.