ITCS early registration deadline
Message from Costis, Yael, and Vinod:
ITCS is back in the east coast, and will be at MIT from January 1114, 2018. As you know, ITCS is a conference that is unique in many respects: it’s a conference that emphasizes dialog and discussion among all subareas of TCS, facilitating it with a single track structure and “chair rants” providing the context for each session. Submissions, refereeing and presentations emphasize the “I” in ITCS: new concepts and models, new lines of inquiry, new techniques or novel use of existing techniques, and new connections between areas.
All in all, great fun! This year, ITCS will run for four full days with lots of activities. Tickets are going fast: the deadline for early registration and hotel block are both December 28, 2017.
A great tradition at ITCS is the “graduating bits” session, where graduating PhD students and postdocs give brief overviews of their research in advance of going out on the job market. If you fit the description, you should sign up here.
Following the success of the poster session at ITCS’17 and STOC’18, we will have one too, at the Marriott the first evening of the conference. To sign up, go here.
Sam Hopkins’s 6 part learning via SoS series
(I’m a non native speaker – is it Hopkins’ or Hopkins’s? –Boaz)
Sam Hopkins just completed a heroic 6 part blog post sequence on using the Sum of Squares algorithm for unsupervised learning.
The goal of unsupervised learning is to recover the underlying structure of a distribution given samples sampled from $\mathcal{D}$. This is phrased as positing a model for the distribution as having the form where is a parameter, and the goal is to recover the parameter from the examples. We can consider the informationtheoretic or computationally unbounded setting where the question is of identifiability – is there enough data to recover (an approximation of) the parameter, and the more realistic setting where we are computationally bounded and the question becomes one of recovery – can we efficiently recover the parameter from the data.
Theoretical computer scientists are used to the setting where every problem can be solved if you have enough computational power, and so intuitively would think that identifiability and recovery have nothing to do with each other, and that the former would be a much easier task than the latter. Indeed this is often the case, but Sam discussed some important cases where we can (almost) automatically transform a proof of identifiability into an efficient algorithm for recovery.
While reading these 6 (not too long) blog posts might take an afternoon, it is an afternoon you would learn something in, and I highly recommend this series, especially for students who might be interested in pursuing questions in the theory of machine learning or the applications of semidefinite programming.
The series is as follows:
 In part I Sam describes the general model of unsupervised learning and focuses on the classical problem (dating at least as far back as the 19th century) of learning a Gaussian mixture model , but for which significant progress was just made by Sam and others. He wisely focuses on the one dimensional case, and gives a proof of identifiability for this case.
 In part II Sam introduces the Sum of Squares proof system, and starts on an SoS proof (and statement) of his identifiability proof from the first Part. He completes this proof of identifiability in the (relatively short) part III.
 In the part IV, Sam completes the transformation of this identifiability proof into an algorithm for the one dimensional Gaussian mixture model.
 The above already gives a complete description of how to transform and identifiability proof to an algorithm, but of course we really care about the high dimensional case. In Part V Sam generalizes the identifiability proof to higher dimensions,
 In Part VI Sam completes the transformation of the higher dimensional identifiability proof to an algorithm, and also includes an overview of other papers and resources on this general area.
While I realize that today’s social media is trending towards 140 characters, I hope that some people still have the attention span for a longer exposition, and I do believe those that read it will find it worthwhile.
Thanks so much Sam!
Clustering and Sum of Squares Proofs, Part 6
This is the 6th and final part of a series on clustering, Gaussian mixtures, and Sum of Squares (SoS) proofs. If you have not read them yet, I recommend starting with Part 1, Part 2, Part 3, Part 4, and Part 5. Also, if you find errors please mention them in the comments (or otherwise get in touch with me) and I will fix them ASAP.
Did you find this series helpful? Unhelpful? What could be done better? If there were to be another tutorial series on a Sum of Squaresrelated topic, what would you like to see? Let me know in the comments!
Last time we developed our highdimensional clustering algorithm for Gaussian mixtures. In this post we will make our SoS identifiability proofs highdimensional. In what is hopefully a familiar pattern by now, these identifiability proofs will also amount to an analysis of the clustering algorithm from part 5. At the end of this post is a modest discussion of some of the literature on the SoS proofstoalgorithms method we have developed in this series.
Setting
We decided to remember the following properties of a collection of samples from a dimensional Gaussian mixture model.
(1) break up into clusters which partition , and each has size exactly .
(2) Each cluster has bounded moments, and this has a small certificate: for some , if is the mean of the th cluster,
.
(3) The means are separated: if then .
Our identifiability proof follows the template we have laid out in the nonSoS proof from part 1 and the onedimensional SoS proof from later parts. The first thing to do is prove a key fact about a pair of overlapping clusters with bounded moments.
Fact 3
The next fact is the highdimensional analogue of Fact 2. (We are not going to prove a highdimensional analogue of Fact 1; the reader should at this point have all the tools to work it out for themselves.) We remind the reader of the key family polynomial inequalities.
Given a collection of points , let be the following set of polynomial inequalities in indeterminates :
for all
where as usual . As before, for a subset , we use the notation .
Fact 3.
Let . Let have ; let be its mean. Let be a power of . Suppose satisfies.
Then
The main difference between the conclusions of Fact 3 and Fact 2 is that both sides of the inequality are multiplied by , as compared to Fact 2. As we will see, this is because an additional dependence on the vectorvalued polynomial is introduced by the need to project the highdimensional vectors onto the line . We have already tackled a situation where an SoSprovable inequality seemed to require cancelling terms of left and right in order to be useful (i.e. when we used Fact 2 to prove Lemma 2), and similar ideas will work here.
The main innovation in proving Fact 3 is the use of the th moment inequality in . Other than that, we follow the proof of Fact 2 almost line by line. The main proposition needed to use the th moment inequality is:
Proposition. .
Proof of Proposition.
Expanding the polynomial we get
where is a multiindex over and “even” means that every index in occurs with even multiplicity. (Other terms vanish by symmetry of .) Since is always even in the sum, the monomial is a square, and by standard properties of Gaussian moments. Hence,
.
QED.
Proof of Fact 3.
As usual, we write things out in terms of , then apply Holder’s inequality and the triangle inequality. First of all,
By SoS Holder’s inequality, we get
By the same reasoning as in Fact 2, using and we get
By our usual squaring and use of , we also get
(We want both sides to be squared so that we are set up to eventually use SoS CauchySchwarz.) We are left with the last two terms, which are th moments in the direction . If we knew
and similarly
then we would be done. We start with the second inequality. We write the polynomial on the LHS as
Squaring again as usual, it would be enough to bound both and . For the former, using the Proposition above we get
.
For the latter, notice that
where is the matrix
.
Hence by SoS CauchySchwarz, we get
Putting these together, we get
,
the second of the inequalities we wanted. Proving the first one is similar, using the hypothesis
in place of . QED.
Lemma 3
The last thing is to use Fact 3 to prove Lemma 3, our highdimensional SoS identifiability lemma. Predictably, it is almost identical to our previous proof of Lemma 2 using Fact 2.
Lemma 3.
Let . Let be a partition of into pieces of size such that for each , the collection of vectors obeys the following moment bound:.
where is the average and is some number in which is a power of . Let be such that for every .
Let be indeterminates. Let be the set of equations and inequalities defined above. Thinking of the variables as defining a set via its indicator, let be the formal expression Let be a degree pseudoexpectation which satisfies . Then
Proof of Lemma 3.
Let satisfy . As in Lemmas 1 and 2, we will endeavor to bound for each .
By separation,
where we implicitly used the SoS triangle inequality on each coordinate of the vectors and .
So,
Now we are going to bound . By symmetry the same argument will apply to the same expression with and exchanged.
We apply Fact 3:
Then we use pseudoexpectation CauchySchwarz to get
.
Putting these two together and canceling we get
.
Also, clearly . So we get
We started out with the goal of bounding . We have found that
Applying pseudoexpectation Holder’s inequality, we find that
.
Rearranging things, we get
Now proceeding as in the proof of Lemma 2, we know
so
QED.
Remark: We cheated ever so slightly in this proof. First of all, we did not state a version of pseudoexpectation Holder’s which allows the exponent , just one which allows . The correct version can be found in Lemma A.4 of this paper. That inequality will work only when is large enough; I think suffices. To handle smaller probably one must remove the square from both sides of Fact 3, which will require a hypothesis which does not use the squared Frobenious norm. This is possible; see e.g. my paper with Jerry Li.
Putting Things Together
The conclusion of Lemma 3 is almost identical to the conclusion of Lemma 2, and so the rest of the analysis of the highdimensional clustering algorithm proceeds exactly as in the onedimensional case. At the end, to show that the clustering algorithm works with high probability to cluster samples from a separated Gaussian mixture model, one uses straightforward concentration of measure to show that if are enough samples from a separated mixture model, then the satisfy the hypotheses of Lemma 3 with high probability. This concludes our “proof” of the main theorem from way back in part 1.
Related literature
The reader interested in further applications of the Sum of Squares method to unsupervised learning problems may consult some of the following works.
 [Barak, Kelner, Steurer] on dictionary learning
 [Potechin, Steurer] on tensor completion
 [Ge, Ma] on random overcomplete tensors
Though we have not seen it in these posts, often the SoS method overlaps with questions about tensor decomposition. For some examples in this direction, see the [Barak, Kelner, Steurer] dictionary learning paper above, as well as
 [Ma, Shi, Steurer] on tensor decomposition
The SoS method can often be used to design algorithms which have more practical running times than the large SDPs we have discussed here. (This often requires further ideas, to avoid solving large semidefinite programs.) See e.g.:
 [Hopkins, Schramm, Shi, Steurer] on extracting spectral algorithms (rather than SDPbased algorithms) from SoS proofs
 [Schramm, Steurer] developing a sophisticated spectral method for dictionary learning via SoS proofs
 [Hopkins, Kothari, Potechin, Raghavendra, Schramm, Steurer] with a metatheorem on when it is possible to extract spectral algorithms from SoS proofs
Another common tool in constructing SoS proofs for unsupervised learning problems which we did not see here are concentration bounds for random matrices whose entries are lowdegree polynomials in independent random variables. For some examples along these lines, see
 [Hopkins, Shi, Steurer] on tensor principal component analysis
 [Rao, Raghavendra, Schramm] on random constraint satisfaction problems
as well as several of the previous papers.
Clustering and Sum of Squares Proofs, Part 5
This is part 5 of a continuing series on clustering, Gaussian mixtures, and Sum of Squares (SoS) proofs. If you have not read them yet, I recommend starting with Part 1, Part 2, Part 3, and Part 4. Also, if you find errors please mention them in the comments (or otherwise get in touch with me) and I will fix them ASAP.
Last time we finished our algorithm design and analysis for clustering onedimensional Gaussian mixtures. Clustering points on isn’t much of a challenge. In this post we will finally move to the highdimensional setting. We will see that most of the ideas and arguments so far carry over nearly unchanged.
In keeping with the method we are advocating throughout the posts, the first thing to do is return to the nonSoS cluster identifiability proof from Part 1 and see how to generalize it to collections of points in dimension . We encourage the reader to review that proof.
Generalizing the nonSoS Identifiability Proof
Our first step in designing that proof was to correctly choose a property of a collection of samples from a Gaussian mixture which we would rely on for identifiability. The property we chose was that the points break into clusters of equal size so that each cluster has bounded empirical th moments and the means of the clusters are separated.
Here is our first attempt at a highdimensional generalization: break into clusters of equal size such that
(1) for each cluster and ,
where is the empirical mean of cluster , and
(2) those means are separated: for .
The first property says that every onedimensional projection of every cluster has Gaussian th moments. The second should be familiar: we just replaced distance on the line with distance in .
The main steps in our onedimensional nonSoS identifiability proofs were Fact 1 and Lemma 1. We will give an informal discussion on their highdimensional generalizations; for the sake of brevity we will skip a formal nonSoS identifiability proof this time and go right to the SoS proof.
The key idea is: for any pair of sets such that and satisfy the empirical th moment bound (2) with respect to empirical means and respectively, if , then by the onedimensional projections
are collections of numbers in which satisfy the hypotheses of our onedimensional identifiability arguments. All we did was choose the right onedimensional projection of the highdimensional points to capture the separation between and .
(The reader is encouraged to work this out for themselves; it is easiest shift all the points so that without loss of generality .)
Obstacles to Highdimensional SoS Identifiability
We are going to face two main difficulties in turning the highdimensional nonSoS identifiability proofs into SoS proofs.
(1) The onedimensional projections above have in the denominator, which is not a lowdegree polynomial. This is easy to handle, and we have seen similar things before: we will just clear denominators of all inequalities in the proofs, and raise both sides to a highenough power that we get polynomials.
(2) The highdimensional th moment bound has a “for all ” quantification. That is, if are indeterminates as in our onedimensional proof, to be interpreted as the indicators for membership in a candidate cluster , we would like to enforce
.
Because of the , this is not a polynomial inequality in . This turns out to be a serious problem, and it will require us to strengthen our assumptions about the points .
In order for the SoS algorithm to successfully cluster , it needs to certify that each of the clusters it produces satisfies the th empirical moment property. Exactly why this is so, and whether it would also be true for nonSoS algorithms, is an interesting topic for discussion. But, for the algorithm to succeed, in particular a short certificate of the above inequality must exist! It is probably not true that such a certificate exists for an arbitrary collection of points in satisfying the th empirical moment bound. Thus, we will add the existence of such a certificate as an assumption on our clusters.
When are sufficientlymany samples from a dimensional Gaussian, the following matrix inequality is a short certificate of the th empirical moment property:
where the norm is Frobenious norm (spectral norm would have been sufficient but the inequality is easier to verify with Frobenious norm instead, and this just requires taking a few more samples). This inequality says that the empirical th moment matrix of is close to its expectation in Frobenious norm. It certifies the th moment bound, because for any , we would have
by analyzing the quadratic forms of the empirical and true th moment matrices at the vector .
In our highdimensional SoS identifiability proof, we will remember the following things about the samples from the underlying Gaussian mixture.
 break into clusters , each of size , so that if is the empirical mean of the th cluster, if , and
 For each cluster :
.
Algorithm for HighDimensional Clustering
Now we are prepared to describe our highdimensional algorithm for clustering Gaussian mixtures. For variety’s sake, this time we are going to describe the algorithm before the identifiability proof. We will finish up the highdimensional identifiability proof, and hence the analysis of the following algorithm, in the next post, which will be the last in this series.
Given a collection of points , let be the following set of polynomial inequalities in indeterminates :
for all
where as usual .
The algorithm is: given , find a degree pseudoexpectation of minimal satisfying . Run the rounding procedure from the onedimensional algorithm on .
Clustering and Sum of Squares Proofs, Part 4
This is part 4 of a continuing series on clustering, Gaussian mixtures, and Sum of Squares (SoS) proofs. If you have not read them yet, I recommend starting with Part 1, Part 2, and Part 3. Also, if you find errors please mention them in the comments (or otherwise get in touch with me) and I will fix them ASAP.
Last time we finished our SoS identifiability proof for onedimensional Gaussian mixtures. In this post, we are going to turn it into an algorithm. In Part 3 we proved Lemma 2, which we restate here.
Lemma 2.
Let . Let be a partition of into pieces of size such that for each , the collection of numbers obeys the following moment bound:where is the average and is a power of in . Let be such that for every .
Let be indeterminates. Let be the following set of equations and inequalities.
As before is the polynomial . Thinking of the variables as defining a set via its indicator, let be the formal expression
Let be a degree pseudoexpectation which satisfies . Then
We will we design a convex program to exploit the SoS identifiability proof, and in particular Lemma 2. Then we describe a (very simple) rounding procedure and analyze it, which will complete our description and analysis of the onedimensional algorithm.
Let’s look at the hypothesis of Lemma 2. It asks for a pseudoexpectation of degree which satisfies the inequalities . First of all, note that the inequalities depend only on the vectors to be clustered, and in particular not on the hidden partition , so they are fair game to use in our algorithm. Second, it is not too hard to check that the set of pseudoexpectations satisfying is convex, and in fact the feasible region of a semidefinite program with variables!
It is actually possible to design a rounding algorithm which takes any pseudoexpectation satisfying and produces a cluster, up to about a fraction of misclassified points. Then the natural approach to design an algorithm to find all the clusters is to iterate:
(1) find such a pseudoexpectation via semidefinite programming
(2) round to find a cluster
(3) remove all the points in from , go to (1).
This is a viable algorithm, but analyzing it is a little painful because misclassifications from early rounds of the rounding algorithm must be taken into account when analyzing later rounds, and in particular a slightly stronger version of Lemma 2 is needed, to allow some error from early misclassifications.
We are going to avoid this pain by imposing some more structure on the pseudoexpectation our algorithm eventually rounds, to enable our rounding scheme to recover all the clusters without resolving a convex program. This is not possible if one is only promised a pseudoexpectation which satisfies : observe, for example, that one can choose the pseudodistribution to be a probability distribution supported on one point , the indicator of cluster . This particular is easy to round to extract , but contains no information about the remaining clusters .
We are going to use a trick reminiscent of entropy maximization to ensure that the pseudoexpectation we eventually round contains information about all the clusters . Our convex program will be:
where is the Frobenius norm of the matrix .
It may not be so obvious why is a good thing to minimize, or what it has to do with entropy maximization. We offer the following interpretation. Suppose that instead of pseudodistributions, we were able to minimize over all which are supported on vectors which are the indicators of the clusters . Such a distribution is specified by nonnegative for which sum to , and the Frobenius norm is given by
where we have used orthogonality if . Since all the clusters have size , we have , and we have , where is the norm, whose square is the collision probability, of . This collision probability is minimized when is uniform.
We can analyze our convex program via the following corollary of Lemma 2.
Corollary 1.
Let and be as in Lemma 2. Let be the degree pseudoexpectation solvingLet be the uniform distribution over vectors where is the indicator of cluster . Then
where is the Frobenius norm.
Proof of Corollary 1.
The uniform distribution over is a feasible solution to the convex program with , by the calculation preceding the corollary. So if is the minimizer, we know .
We expand the norm:
To bound the last term we use Lemma 2. In the notation of that Lemma,
Remember that . Putting these together, we get
QED.
We are basically done with the algorithm now. Observe that the matrix contains all the information about the clusters . In fact the clusters can just be read off of the rows of .
Once one has in hand a matrix which is close in Frobenius norm to , extracting the clusters is still a matter of reading them off of the rows of the matrix (choosing rows at random to avoid hitting one of the small number of rows which could be wildly far from their sisters in ).
We will prove the following fact at the end of this post.
Fact: rounding.
Let be a partition of into parts of size . Let be the indicator matrix for samecluster membership. That is, if and are in the same cluster . Suppose $M \in \mathbb{R}^{n \times n}$ satisfies .There is a polynomialtime algorithm which takes and with probability at least produces a partition of into clusters of size such that, up to a permutation of ,
Putting things together for onedimensional Gaussian mixtures
Now we sketch a proof of Theorem 1 in the case . Our algorithm is: given , solve
then apply the rounding algorithm from the rounding fact to and output the resulting partition.
If the vectors satisfy the hypothesis of Lemma 2, then by Corollary 1, we know
where is the uniform distribution over indicators for clusters . Hence the rounding algorithm produces a partition of such that
Since a standard Gaussian has , elementary concentration shows that the vectors satisfy the hypotheses of Lemma 1 with probability so long as .
Rounding algorithm
The last thing to do in this post is prove the rounding algorithm Fact. This has little to do with SoS; the algorithm is elementary and combinatorial. We provide it for completeness.
The setting is: there is a partition of into parts of size . Let be the indicator matrix for cluster membership; i.e. if and only if and are in the same cluster . Given a matrix such that , the goal is to recover a partition of which is close to up to a permutation of .
Let be the th row of and similarly for . Let be a parameter we will set later.
The algorithm is:
(1) Let be the set of active indices.
(2) Pick uniformly.
(3) Let be those indices for which .
(4) Add to the list of clusters and let $\mathcal{I} := \mathcal{I} \setminus S$.
(5) If , go to (2).
(6) (postprocess) Assign remaining indices to clusters arbitrarily, then move indices arbitrarily from larger clusters to smaller ones until all clusters have size .
Fact: rounding.
If then with probability at least the rounding algorithm outputs disjoint clusters , each of size , such that up to a permutation of , .
Proof.
Call an index good if . An index is bad if it is not good. By hypothesis . Each bad index contributes at least to the left side and , so there are at most bad indices.
If are good indices and both are in the same cluster , then if the algorithm chooses , the resulting cluster will contain . If , then also if is good but is in some other cluster , the cluster formed upon choosing will not contain . Thus if the algorithm never chooses a bad index, before postprocessing the clusters it outputs will (up to a global permutation of ) satisfy that contains all the good indices in . Hence in this case only bad indices can be misclassified, so the postprocessing step moves at most indices, and in the end the cluster again errs from on at most indices.
Consider implementing the algorithm by drawing a list of indices before seeing (i.e. obliviously), then when the algorithm requires random index we give it the next index in our list which is in (and halt with no output if no such index exists). It’s not hard to see that this implementation fails only with probability at most . Furthermore, by a union bound the list contains a bad index only with probability . Choosing thus completes the proof. QED.
Clustering and Sum of Squares Proofs, Part 3
This is part 3 of a continuing series on clustering, Gaussian mixtures, and Sum of Squares (SoS) proofs. If you have not read them yet, I recommend starting with Part 1 and Part 2. Also, if you find errors please mention them in the comments (or otherwise get in touch with me) and I will fix them ASAP.
The Story So Far
Let’s have a brief recap. We are designing an algorithm to cluster samples from Gaussian mixture models on . Our plan is to do this by turning a simple identifiability proof into an algorithm. For us, “simple” means that the proof is captured by the low degree Sum of Squares (SoS) proof system.
We have so far addressed only the case (which will remain true in this post). In part 1 we designed our identifiability proof, not yet trying to formally capture it in SoS. The proof was simple in the sense that it used only the triangle inequality and Holder’s inequality. In part 2 we defined SoS proofs formally, and stated and proved an SoS version of one of the key facts in the identifiability proof (Fact 2).
In this post we are going to finish up our SoS identifiability proof. In the next post, we will see how to transform the identifiability proof into an algorithm.
Setting
We recall our setting formally. Although our eventual goal is to cluster samples sampled from a mixture of Gaussians, we decided to remember only a few properties of such a collection of samples, which will hold with high probability.
The properties are:
(1) They break up into clusters of equal size, , such that for some , each cluster obeys the empirical moment bound,
where is the empirical mean of the cluster , and
(2) Those means are separated: .
The main statement of cluster identifiability was Lemma 1, which we restate for convenience here.
Lemma 1. Let . Let be a partition of into pieces of size such that for each , the collection of numbers obeys the following moment bound:
where is the average and is some number in . Let be such that for every . Suppose is large enough that .
Let have size and be such that obey the same momentboundedness property:
for the same , where is the mean . Then there exists an such that
for some universal constant .
Our main goal in this post is to state and prove an SoS version of Lemma 1. We have already proved the following Fact 2, an SoS analogue of Fact 1 which we used to prove Lemma 1.
Fact 2. Let . Let have ; let be its mean. Let be a power of . Suppose satisfies
Let be indeterminates. Let be the following set of equations and inequalities.
Then
Remaining obstacles to an SoS version of Lemma 1
We are going to face a couple of problems.
(1) The statement and proof of Lemma 1 are not sufficiently symmetric for our purposes — it is hard to phrase things like “there exists a cluster such that…” as statements directly about polynomials. We will handle this by giving more symmetric version of Lemma 1, with a more symmetric proof.
(2) Our proof of Lemma 1 uses the conclusion of Fact 1 in the form
whereas Fact 2 concludes something slightly different:
The difference in question is that the polynomials in Fact 2 are degree , and appears on both sides of the inequality. If we were not worried about SoS proofs, we could just cancel terms in the second inequality and take th roots to obtain the first, but these operations are not necessarily allowed by the SoS proof system.
One route to handling this would be to state and prove a version of Lemma 1 which concerns only degree . This is probably possible but definitely inconvenient. Instead we will exhibit a common approach to situations where it would be useful to cancel terms and take roots but the SoS proof system doesn’t quite allow it: we will work simultaneously with SoS proofs and with their dual objects, pseudodistributions.
We will tackle issues (1) and (2) in turn, starting with the (a)symmetry issue.
Lemma 1 reformulated: maintaining symmetry
We pause here to record an alternative version of Lemma 1, with an alternative proof. This second version is conceptually the same as the one we gave in part 1, but it avoids breaking the symmetry among the clusters , whereas this was done at the very beginning of the first proof, by choosing the ordering of the clusters by . Maintaining this symmetry requires a slight reformulation of the proof, but will eventually make it easier to phrase the proof in the Sum of Squares proof system. In this proof we will also avoid the assumption , however, we will pay a factor of rather than in the final bound.
Alternative version of Lemma 1.
Let .
Let be a partition of into pieces of size such that for each , the collection of numbers obeys the following moment bound:where is the average and is some number in . Let be such that for every .
Let have size and be such that obey the same momentboundedness property:
.
for the same , where . Then
We remark on the conclusion of this alternative version of Lemma 1. Notice that are nonnegative numbers which sum to . The conclusion of the lemma is that for . Since the sum of their squares is at least , one obtains
matching the conclusion of our first version of Lemma 1 up to an extra factor of .
Proof of alternative version of Lemma 1.
Let again have size with mean and th moment bound . Since partition ,
We will endeavor to bound for every pair . Since ,
Certainly and similarly for , so this is at most
Using Fact 1, this in turn is at most . So, we obtained
for every .
Putting this together with our first bound on , we get
QED.
Now that we have resolved the asymmetry issue in our earlier version of Lemma 1, it is time to move on to pseudodistributions, the dual objects of SoS proofs, so that we can tackle the last remaining hurdles to proving an SoS version of Lemma 1.
Pseudodistributions and duality
Pseudodistributions are the convex duals of SoS proofs. As with SoS proofs, there are several expositions covering elementary definitions and results in detail (e.g. the lecture notes of Barak and Steurer, here and here). We will define what we need to keep the tutorial selfcontained but refer the reader elsewhere for further discussion. Here we follow the exposition in those lecture notes.
As usual, let be some indeterminates. For a finitelysupported function and a function , define
If defines a probability distribution, then is the operator sending a function to its expectation under .
A finitelysupported is a degree pseudodistribution if
(1)
(2) for every polynomial of degree at most .
When is clear from context, we usually suppress it and write . Furthermore, if is an operator and for some pseudodistribution , we often abuse terminology and call a pseudoexpectation.
If is a family of polynomial inequalities and is a degree pseudodistribution, we say satisfies if for every and such that one has
We are not going to rehash the basic duality theory of SoS proofs and pseudodistributions here, but we will need the following basic fact, which is easy to prove from the definitions.
Fact: weak soundness of SoS proofs.
Suppose and that is a degree pseudodistribution which satisfies . Then for every SoS polynomial , if $\deg h + \ell \leq d$ then .
We call this “weak soundness” because somewhat stronger statements are available, which more readily allow several SoS proofs to be composed. See Barak and Steurer’s notes for more.
The following fact exemplifies what we mean in the claim that pseudodistributions help make up for the inflexibility of SoS proofs to cancel terms in inequalities.
Fact: pseudoexpectation CauchySchwarz.
Let be a degree pseudoexpectation on indeterminates . Let and be polynomials of degree at most . ThenAs a consequence, if has degree and is a power of , by induction
Proof of pseudoexpectation CauchySchwarz.
For variety, we will do this proof in the language of matrices rather than linear operators. Let be the matrix indexed by monomials among of degree at most , with entries . If is a polynomial of degree at most , we can think of as a vector indexed by monomials (whose entries are the coefficients of ) such that . Hence,
QED.
We will want a second, similar fact.
Fact: pseudoexpectation Holder’s.
Let be a degree sum of squares polynomial, , and a degree pseudoexpectation. Then
The proof of pseudoexpectation Holder’s is similar to several we have already seen; it can be found as Lemma A.4 in this paper by Barak, Kelner, and Steurer.
Lemma 2: an SoS version of Lemma 1
We are ready to state and prove our SoS version of Lemma 1. The reader is encouraged to compare the statement of Lemma 2 to the alternative version of Lemma 1. The proof will be almost identical to the proof of the alternative version of Lemma 1.
Lemma 2.
Let . Let be a partition of into pieces of size such that for each , the collection of numbers obeys the following moment bound:where is the average and is a power of in . Let be such that for every .
Let be indeterminates. Let be the following set of equations and inequalities.
As before is the polynomial . Thinking of the variables as defining a set via its indicator, let be the formal expression
Let be a degree pseudoexpectation which satisfies . Then
Proof of Lemma 2.
We will endeavor to bound from above for every . Since we want to use the degree polynomials in Fact 2, we get started with
by (repeated) pseudoexpectation CauchySchwarz.
Since and are separated, i.e. , we also have
where the indeterminate is and we have only used the SoS triangle inequality. Hence,
Applying Fact 2 and soundness to the righthand side, we get
Now using that and hence and similarly for , we get
By pseudoexpectation CauchySchwarz
which, combined with the preceding, rearranges to
By pseudoexpectation Holder’s,
All together, we got
Now we no longer have to worry about SoS proofs; we can just cancel the terms on either side of the inequality to get
Putting this together with
finishes the proof. QED.
Clustering and Sum of Squares Proofs, Part 2
This is part 2 of a series on clustering, Gaussian mixtures, and Sum of Squares (SoS) proofs. If you have not read it yet, I recommend starting with Part 1. Also, if you find errors please mention them in the comments (or otherwise get in touch with me) and I will fix them ASAP.
Welcome back.
In the last post, we introduced Gaussian mixture models and the clustering problem for Gaussian mixtures. We described identifiability proofs for unsupervised learning problems. Then we set ourselves some goals:
 Design a simple identifiability proof for clustering in Gaussian mixtures, saying that if are (enough) samples from a dimensional mixture of Gaussians, the groundtruth clustering of the ‘s by which Gaussian they were drawn from is identifiable from the samples.
 Formalize the simplicity of that identifiability proof by showing that it is captured by a formal proof system of restricted power: the Sum of Squares (SoS) proof system.
 Guided by our SoS identifiability proof, design an algorithm for clustering in Gaussian mixtures. (And hence prove Theorem 1 from last time.)
In the last post, we accomplished task (1) in dimensional case. In this post we will get started on task (2), again in the dimensional case.
Recalling from last time, in our identifiability proof we remembered only two things things about our samples :
(1) They break up into clusters of equal size, , such that for some , each cluster obeys the empirical moment bound,
where is the empirical mean of the cluster , and
(2) Those means are separated: .^{1}
The key tool we used in our identifiability proof was Fact 1, which we restate here for convenience.
Fact 1. Let have . Let denote a uniform sample from and similarly for . Let and . Suppose satisfy the th moment bound
Then
We are going to give a Sum of Squares proof of this fact. Or rather: we are going to state a very similar fact, which concerns inequalities among lowdegree polynomials, and give a Sum of Squares proof of that.
We are going to do things in a slightly unusual order, delaying definition of the SoS proof system till we have something concrete in mind to prove in it.
First, because SoS is a proof system to reason about inequalities among lowdegree polynomials, we are going to formulate Fact 2, which is like Fact 1 except it will be explicitly about lowdegree polynomials. Proving Fact 2 will be our goal.
Second, we will define the SoS proof system.
Finally, we will prove this Fact 2 in the lowdegree SoS proof system.
Fact 2: an SoS version of Fact 1
Fact 1 concerns two subsets of . When we used Fact 1 to prove Lemma 1 in part 1, one of those sets was one of the groundtruth clusters among , and one of them was a “candidate” cluster — Lemma 1 showed that the candidate cluster must in fact have been close to one of the true clusters .
We will design a system of polynomial equations whose solutions are in correspondence with candidate clusters . This is probably unavoidably foreign at first. We will offer some attempts at demystifying remarks later on, but for now we will forge ahead.
Let . Let be some indeterminates; they will be the variables in our polynomials. We are going to think of them as indicators of a subset which is a candidate cluster.
First, let’s enforce that is the indicator vector of a set of size . Consider the equations
.
Any solution to these equations over is a indicator of a subset of of size .
The second hypothesis Fact 1 places on a candidate cluster is the th moment bound. Let’s enforce that with a polynomial inequality. First, we need some notation for the empirical mean of . Denote by the polynomial
.
Often we drop the and just write . And now consider the inequality:
.
Belaboring the point somewhat: any solution over to the equations and inequalities we have described would correspond to a subset of of which obeys the th empirical moment bound. (Here we assume that is even, so that .)
Although we don’t have all the definitions we need in place yet, we will go ahead and state Fact 2. We introduce some suggestive notation. If , we define the polynomial
.
Often we just write .
Fact 2. Let . Let have ; let be its mean. Let be a power of . Suppose satisfies
Let be indeterminates. Let be the following set of equations and inequalities.
Then
We have not yet defined the notation . This notation means that that there is a degree SoS proof of the inequality on the righthand side using the axioms — we will define this momentarily.
The purpose of stating Fact 2 now was just to convince the reader that there is a plausible version of Fact 1 which may be stated entirely as inequalities among polynomials. The reader is encouraged to compare the hypotheses and conclusions of Facts 1 and 2 (ignoring this for now).^{2} There are two main differences:
(a) The inequality in the conclusion of Fact 2 is raised to the th power, as compared to the conclusion of Fact 1.
(b) The inequality in the conclusion of Fact 2 seems to have extra factors of on both sides.
Point (a) is needed just to make both sides of the inequality into polynomials in ; otherwise there would be fractional powers. The need for (b) is more subtle, and we will not be able to fully understand it for a while. For now, what we can say is: the inequality
would be true for any which solves , but that might not have an SoS proof.
One last difference is the factor on the righthand side of the conclusion. This is probably not inherent; it arises because we will use an easytoprove but lossy version of the triangle inequality in our SoS proof. In any case, it is dwarfed by the term , so we are not too worried about it.
Enough dancing around these SoS proofs — in order to stop with the handwaiving, we need to set up our proof system.
Sum of Squares Proofs
We cannot go any further without defining the formal proof system we will work in. Since for now we are sticking with a onedimensional setting, things can be little simpler than for the proof system we will need when become highdimensional, but developing the proof system still incurs a little notational burden. That is life.
The Sum of Squares proof system, henceforth “SoS”, is a formal proof system for reasoning about systems of polynomial equations and inequalities over the real numbers. At its heart is the simple fact that if is a polynomial in indeterminates with real coefficients, then for all .
Plenty of elementary expositions on SoS proofs are available (see e.g. SoS on the hypercube, SoS on general domains, and the first several sections of this paper by Ryan O’Donnell and Yuan Zhou.) We will define as much as we need to keep this tutorial selfcontained but for expanded discussion we refer the reader to those resources and references therein.
Let be some indeterminates. Let be some polynomial inequalities in those indeterminates. If is some other polynomial with real coefficients, it may be the case that for any satisfying , also ; we would say that implies .
The key concept for us will be that implies with a sum of squares proof. We say that SoSproves if there exist polynomials for such that
and the polynomials are sums of squares—i.e. each of them has the form for some polynomials . Notice that if this equation holds, for any which satisfies the inequalities the righthand side must be nonnegative, so is also nonnegative.
The polynomials form an SoS proof that implies . If are all at most , we say that the proof has degree , and we write
.
When we often just write .
We commonly use a few shorthand notations.
 We write , by which we mean .
 We include polynomial equations such as in , by which we mean that contains both and .
 We write , by which we mean that both and .
Although the definition suggests something static — there is a fixed collection of polynomials forming an SoS proof — in practice we treat SoS proofs as dynamic objects, building them line by line much as we would any other proof. We are going to see an example of this very soon when we prove Fact 2.
It is time to begin to make good on the promise from the last section that we would get substantial mileage out of proving identifiability of mixtures of Gaussians using only simple inequalities. While it will take us several sections to completely make good on our promise, we can begin by giving SoS versions of the triangle and Holder’s inequalities we used in our identifiability proof. We will prove one of these now to give the reader a sense of how such arguments go; since there is usually not much novelty in SoS proofs of such basic inequalities we will defer others till later.
We would like to emphasize that the following SoS proofs themselves have nothing to do with mixtures of Gaussians; instead they are part of a growing problemindependent toolkit of basic inequalities useful in designing SoS proofs of more interesting mathematical statements. The fact that one can have such a problemindependent toolkit is in part what makes the proofstoalgorithms method so broadly useful.
SoS triangle inequality
We will start with a fairly weak SoS triangle inequality, that will suffice for our needs.
Much more sophisticated versions of this inequality are possible which allow various norms and do not lose the factor we do here.
Fact: SoS triangle inequality.
Let be indeterminates. Let be a power of . Then
To prove the SoS triangle inequality we will want the following basic observation about composability of SoS proofs. (This is really a special case of a more general composibility result.)
Proposition: squaring SoS proofs.
Suppose and are sums of squares. Then .
Proof of proposition.
By hypothesis, is a sum of squares polynomial. Now,
so it is a product of sum of squares polynomials, and hence itself a sum of squares.
QED.
Proof of SoS triangle inequality.
We start with the case . In this case, we have . We claim that , since the polynomial is a square. Hence, we find
Now to prove the general case, we proceed by induction. We may suppose
By the proposition, this implies . Now we can apply the base case again to for and to complete the argument.
QED.
SoS Holder’s inequality
Holder’s inequality poses a quandary for SoS proofs, because of the nonintegral exponents in most norms (hence such norms do not naturally correspond to polynomials). Consequently, there are many SoS versions of Holder’s inequality in the literature, choosing various ways to handle this nonintegrality. The version we present here will be most useful for our mixtures of Gaussians proof. We will address the nonintegral powers issue by imposing polynomial inequalities requiring that some of the underlying variables be Boolean.
Since the proof of this SoS Holder’s inequality proceeds via a similar induction to the one we used for the SoS triangle inequality we just proved, we defer it to the end of this post.
SoS Holder’s inequality.
Let be indeterminates and let be the collection of equationsNote that the only solutions to are .
Let be polynomials of degree at most . Let be a power of . Then
and
SoS Boolean Inequalities
We will also want one more SoS inequality for our proof of Fact 1. In linear programming relaxations of Boolean problems, it is common to replace an integrality constraint with the linear inequalities . SoS can derive the latter inequalities.
Fact: SoS Boolean Inequalities
.
Proof of Fact.
For the inequality , just use the axiom . For the inequality , write
.
QED.
Proof of Fact 2
Without further ado, we will prove Fact 2 by lifting the proof of Fact 1 into the SoS proof system. Though slightly notationally cumbersome, the proof follows that of Fact 1 nearly line by line—the reader is encouraged to compare the two proofs.
Proof of Fact 2.
We write out what we want to bound in terms of , then apply Holder’s inequality and the triangle inequality.
We deploy our SoS Holder’s inequality to obtain
Next we can use our equations to conclude that in fact
The polynomial is a sum of squares, as is via our SoS triangle inequality; applying this with and we obtain
We can add the sum of squares to obtain
Using the equations , which, as in the SoS Boolean inequalities fact can be used to prove , we obtain
Finally using and , we get
Last of all, using to simplify the term ,
The fact follows by rearranging. QED.
SoS proof of Holder’s inequality
The last thing we haven’t proved is our SoS Holder’s inequality. We will need an SoS CauchySchwarz inequality to prove it.
SoS CauchySchwarz.
Let be indeterminates. Then
Proof of SoS CauchySchwarz.
It is not hard to check that
which is a sum of squares. QED.
Proof of SoS Holder’s inequality.
We start with the case . Using SoS CauchySchwarz, we obtain
This follows from our SoS CauchySchwarz inequality by substituting for and for ; we proved a fact about a sum of squares polynomial in which implies a corresponding fact about a sum of squares in variables and . The latter in turn is a sum of squares in .
To finish the first half of the case, we just need to replace with on the righthand side. By adding the polynomials via the equations , we obtain
To establish the second inequality for the base case, we start by again adding multiples of to get
Then the inequality follows from CauchySchwarz and again adding some multiples of .
Now it’s time for the induction step. We can assume
By again adding multiples of , we obtain
Now both sides are sums of squares. So, by squaring, we find
The proof is finished by applying CauchySchwarz to the last term and cleaning up by adding multiples of as necessary. QED.
Looking Ahead
In the next post, we will use Fact 2 to deduce an SoS version of Lemma 1 (from part 1). Subsequently, we will finish designing an algorithm for onedimensional clustering, proving Theorem 1 (from part 1) in the onedimensional case. Then we will get high dimensional.
Footnotes

Suspicious readers may note that our original Gaussian mixture model assumed that the population means are separated. Because we will draw enough samples that the empirical mean of each cluster is very close to the true (population) mean, this difference can be ignored for now and is easy to handle when we put everything together to analyze our final algorithm.

To make them look even more alike of course we could have introduced notations like , but for concreteness we are keeping the variables at least a little explicit.