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2.3 Inequalities for matrix perturbation We will require some basic results on matrix perturbations. For this, we adopt the following notation: Let [MATH] be a Hermitian matrix. Then [MATH] denote its singular values in order. |
Lemma 17 [Weyl’s inequality] Let [MATH] be real symmetric matrices. Then for any [MATH] [EQUATION] Fact 18 Sch92 Let [MATH] be two psd matrices. Let [MATH] . Then, |
[EQUATION] Fact 19 Ste73 Let [MATH] be two psd matrices. Let [MATH] and [MATH] . Then, [EQUATION] Combining these two facts, we have the following corollary. |
Corollary 20 Let [MATH] and let [MATH] be a psd matrix such that [MATH] . Let [MATH] be real symmetric that [MATH] for [MATH] . Then, |
[MATH] Proof. We first apply Fact 18 to obtain that [EQUATION] Observe that [MATH] . Since [MATH] and [MATH] , we apply Fact 19 to obtain that |
[EQUATION] This finishes the proof. Algorithm to test [MATH] -juntas In this section, we will prove the following theorem. Theorem 21 |
There is an algorithm Test-linear-junta which has the following guarantee: Given oracle access to [MATH] , rank parameter [MATH] , surface area parameter [MATH] and error parameter [MATH] , it makes [MATH] queries and |
1. If [MATH] is a linear [MATH] -junta with [MATH] , then the algorithm outputs yes with probability at least [MATH] 2. If [MATH] is [MATH] -far from any linear [MATH] -junta [MATH] with [MATH] , then the algorithm outputs no with probability at least [MATH] |
Remark 22 A convention that we shall adopt (to avoid proliferation of parameters) is to sometimes ignore the confidence parameter of the testing algorithm. Typically, whenever we can estimate a parameter within [MATH] with [MATH] queries with confidence [MATH] , we can do the usual “median trick” and get the same accur... |
The algorithm Test-linear-junta is described in Figure . The algorithm invokes two different subroutines, Test-surface-area and Test-rank whose guarantees we state now. To do this, we first define the notion of [MATH] smooth function. |
Definition 23 A function [MATH] is said to be [MATH] -smooth if there is a function [MATH] such that [MATH] and [MATH] In other words, a function [MATH] is [MATH] smooth if [MATH] is [MATH] -close to some other function [MATH] (in [MATH] distance) and [MATH] has surface area which is essentially bounded by [MATH] . Wit... |
(due to Neeman Nee14 ). Theorem 24 There is an algorithm Test-surface-area which given oracle access to a function [MATH] and error parameter [MATH] makes [MATH] queries and has the following guarantee: |
1. If [MATH] is a function with surface area at most [MATH] , then the algorithm outputs yes with probability at least [MATH] 2. |
Any function [MATH] which passes the test with probability [MATH] is [MATH] -smooth. Next, we state the guarantee of the routine Test-rank |
Lemma 25 The routine Test-rank has a query complexity of [MATH] . Further, we have 1. If the function [MATH] is a linear- [MATH] -junta, then the algorithm Test-rank outputs yes with probability [MATH] |
2. If [MATH] is a [MATH] -smooth function which is [MATH] -far from a linear [MATH] -junta, then the algorithm Test-rank outputs no with probability [MATH] |
In order to prove Theorem 21 , we will need the following claim which shows that property of closeness to a linear [MATH] -junta and closeness to a smooth function can be certified using a single function. |
Lemma 26 For a function [MATH] , suppose that there is a linear [MATH] -junta [MATH] and a function [MATH] of surface area at most [MATH] such that both [MATH] and [MATH] |
are [MATH] -close to [MATH] . Then there is a function [MATH] that is a linear [MATH] -junta and has surface area at most [MATH] , and which is [MATH] -close to [MATH] |
Proof of Theorem 21 If [MATH] is a linear [MATH] -junta with surface area at most [MATH] , then it passes both the tests Test-surface-area as well as Test-rank with probability [MATH] . Thus, any linear [MATH] -junta with surface area at most [MATH] passes with probability at least [MATH] (so as long as [MATH] , the te... |
On the other hand, suppose [MATH] passes Test-linear-junta with probability [MATH] . Then, applying Theorem 24 is [MATH] smooth. In other words, there is a function [MATH] such that [MATH] which is [MATH] -close to [MATH] Further, since [MATH] passes Test-rank with probability [MATH] , Lemma 25 implies that |
[MATH] is [MATH] -close to some linear [MATH] -junta [MATH] . We now apply Lemma 26 to obtain that [MATH] is [MATH] -close to some function [MATH] which is a linear [MATH] -junta and [MATH] . This concludes the proof. |
We now turn to describing the routine Test-rank and prove Lemma 25 Proof of Lemma 25 The bound on the query complexity of Lemma 25 is immediate from the settings of our parameters and query complexity of Lemma 10 |
The first item (i.e., the completeness of Test-rank ) follows from the fact that if [MATH] is a linear [MATH] -junta, [MATH] is also a linear [MATH] -junta. Consequently, [MATH] is a rank- [MATH] matrix. Then, [MATH] has at most [MATH] non-zero singular values. Thus, if [MATH] are the singular values of [MATH] (in orde... |
The proof of the second item (i.e., the soundness of Test-rank ) is more involved. In particular, we can restate the second item as proving the following lemma. |
Lemma 27 Let [MATH] be a [MATH] -smooth function which is [MATH] -far from a linear [MATH] -junta, then the algorithm Test-rank outputs no with probability [MATH] |
The task of proving this lemma shall be the agenda for the rest of this section. In order to prove Lemma 27 , we will need a few preliminary lemmas. The following lemma says that if a function’s gradient is almost always orthogonal to a subspace [MATH] . Then, the function is close to a [MATH] -junta. |
Lemma 28 Let [MATH] (be a [MATH] function) and let [MATH] be a subspace of rank [MATH] and let [MATH] . Let us assume that [MATH] . Then there is a [MATH] -junta [MATH] such that [MATH] |
Proof. Let us rotate the space so that [MATH] Let us now define [MATH] as [EQUATION] Observe that [MATH] is a [MATH] -junta. Now, for every choice [MATH] , consider the function [MATH] as |
[EQUATION] Observe that [MATH] By applying Lemma 13 [EQUATION] Observe that [MATH] . Thus, we get [EQUATION] This finishes the proof. |
For the rest of this section, when we use the value [MATH] , it will bear the same relation as stated in the description of the algorithm Test-rank (see Figure ). |
Proposition 29 Let [MATH] which is [MATH] -smooth. Then, [MATH] Proof. Since [MATH] is [MATH] smooth, we know that there is a function [MATH] such that [MATH] and [MATH] By using the fact that the operator [MATH] is contractive, we have, |
[EQUATION] Next, we use Proposition 16 to get that [MATH] We can now combine these to get [EQUATION] Lemma 30 Let [MATH] be a [MATH] -smooth function which is [MATH] -far from any linear [MATH] -junta. For any subspace [MATH] of co-dimension at most [MATH] |
[EQUATION] Proof. Applying Proposition 29 , we have that [MATH] . By applying Jensen’s inequality, we have [MATH] . Thus, [MATH] is [MATH] -far from any linear [MATH] -junta (in [MATH] distance). Consequently, we can say that for any [MATH] -junta [MATH] [MATH] . By contrapositive of Lemma 28 , we have that |
[EQUATION] Next, observe that Lemma implies that [EQUATION] The second inequality follows immediately from that [MATH] has range bounded between [MATH] . Combining this with ( ), this implies that |
[EQUATION] We are now in a position to finish the proof of Lemma 27 Proof of Lemma 27 Let [MATH] denote [MATH] . As in Figure , consider the matrix [MATH] whose [MATH] entry is [MATH] . Now, consider the matrix [MATH] whose [MATH] column is [MATH] . Then, observe that [MATH] . We would like to analyze the singular valu... |
[EQUATION] Instead of analyzing the non-zero singular values of [MATH] , we will analyze the non-zero singular values of [MATH] . From now on, let us use [MATH] to denote |
[MATH] . Let us define the sequence of stopping times [MATH] as follows: [MATH] and let [MATH] and [MATH] be the eigenspace formed by the top [MATH] eigenvectors of [MATH] . Then, [MATH] is the smallest [MATH] such that |
[MATH] . We now make the following claim. Claim 31 For [MATH] , the top [MATH] singular values of [MATH] are at least [MATH] Proof. |
We will prove this claim by induction. So, assume that the top [MATH] singular values of [MATH] are all at least [MATH] . Now, for [MATH] , let |
[MATH] be the unit vector in the direction of the component of [MATH] orthogonal to [MATH] . Let [MATH] be the linear span of [MATH] and [MATH] . Now, consider any unit vector [MATH] and express it as [MATH] where [MATH] lies in [MATH] and [MATH] is parallel to [MATH] . Next, observe that |
[EQUATION] The first term [MATH] is at least as large as [MATH] and the second term [MATH] is the same as [MATH] . Next, note that |
[EQUATION] Consequently, [EQUATION] Observe that [EQUATION] The first inequality is immediate from the fact that [EQUATION] is a psd matrix. Thus, we obtain that |
[EQUATION] Now, it is clear that [MATH] is a psd matrix. If the singular values of [MATH] are [MATH] , then by Courant Fischer theorem, we have |
[EQUATION] where [MATH] is the set of all [MATH] dimensional subspaces of [MATH] . Thus, by applying ( ) and observing [MATH] , we get |
[EQUATION] This finishes the proof. Now applying Lemma 30 , we have that conditioned on [MATH] [MATH] is a geometric random variable with parameter (at least) [MATH] . From this, it is not difficult to see that with probability at least [MATH] [MATH] Thus, with probability [MATH] , we can assume that the top [MATH] sin... |
Consequently, we get that the top [MATH] singular values of [MATH] are all at least [MATH] . Now, the algorithm computes a matrix [MATH] such that |
[MATH] . By Weyl’s inequality (Lemma 17 ), we get that the top [MATH] singular values of [MATH] are all at least [MATH] . This proves the lemma. |
We finally give the proof of Lemma 26 . The proof relies on the so-called co-area formula. Lemma 32 Let [MATH] be smooth and [MATH] be bounded and measurable. Then |
[EQUATION] Proof of Lemma 26 By Mag12 , there is a smooth function [MATH] with bounded gradient such that [MATH] and [MATH] Let [MATH] be a [MATH] -dimensional subspace for which [MATH] is an [MATH] -junta, and let [MATH] be a standard Gaussian vector on [MATH] . Let [MATH] be the projection operator for subspace [MATH... |
[MATH] . Let [MATH] be uniformly distributed in [MATH] and define [MATH] by [EQUATION] Note that [MATH] is an [MATH] -junta (because [MATH] is an [MATH] -junta). In expectation over [MATH] , the surface area of [MATH] is |
[EQUATION] which by the co-area formula is equal to [EQUATION] In particular, there exists some [MATH] such that the surface area of [MATH] is at most [MATH] |
Next, we will estimate the distance of [MATH] from [MATH] . By the triangle inequality, [MATH] and so [MATH] . On the other hand, Pythagoras’ theorem implies that [MATH] minimizes [MATH] among all [MATH] -juntas; hence, [MATH] |
and so [MATH] Now, [MATH] takes values in [MATH] and so [MATH] On the other hand, the definition of [MATH] ensures that [EQUATION] |
If [MATH] is the probability that [MATH] , it follows that [MATH] and so [EQUATION] By the triangle inequality [MATH] Choosing [MATH] completes the proof. |
Algorithm to find hidden linear invariant structure In this section, we will prove the following main theorem. Theorem 33 Let [MATH] be a linear- [MATH] -junta with surface area [MATH] . Then, there is an algorithm Find-invariant-structure which for any error parameter [MATH] makes [MATH] queries to [MATH] |
and with probability [MATH] outputs (for some [MATH] ) a function [MATH] so that the following holds: there is an orthonormal set of vectors [MATH] such that |
[EQUATION] Further, there is a set [MATH] of orthonormal vectors such that for [MATH] [MATH] and [MATH] is a relevant subspace of [MATH] |
Our algorithm is quite naïve. First, we “identify” – in some implicit sense – the [MATH] -dimensional subspace on which the linear [MATH] -junta acts. We take a fine net of functions defined on that space, and we test them all until we fine the best one. Obviously, this algorithm is not computationally efficient, and i... |
[MATH] -dimensional subspace – that would require a number of queries that depends on [MATH] One consequence of this is that we do not know how to apply an off-the-shelf learning algorithm (such as the one from KOS08 ). |
Definition 34 A set of vectors [MATH] is said to be [MATH] -linearly independent if the following conditions hold: 1. For all [MATH] |
[MATH] 2. For all [MATH] [MATH] Definition 35 For [MATH] and [MATH] , we say that a set of directions [MATH] is [MATH] -linearly independent, if the following holds: For [MATH] , let [MATH] . If for all [MATH] [MATH] |
By Proposition 11 , it is immediate that as long as [MATH] [MATH] . Thus, if [MATH] is [MATH] -linearly independent, then the directions [MATH] are [MATH] linearly independent. |
Lemma 36 The algorithm Test-candidate-direction described in Figure has the following properties: For noise parameter [MATH] , directions [MATH] [MATH] and candidate direction [MATH] (where [MATH] as well as [MATH] meet the requirements described in Figure ), the algorithm satisfies |
1. The query complexity of the algorithm is [MATH] 2. If the Euclidean distance of [MATH] is at least [MATH] from the subspace [MATH] , then the algorithm outputs yes . Conversely, if the algorithm outputs no , then the Euclidean distance must be less than [MATH] |
Proof. The query complexity bound is just immediate from Lemma 10 and plugging in the value of [MATH] from Lemma 57 . To prove the second guarantee, let us use [MATH] to denote [MATH] . Since [MATH] are [MATH] -linearly independent, hence by Lemma 57 , we obtain that there are orthonormal vectors [MATH] (which span [MA... |
[EQUATION] This implies that if we let [MATH] (using [MATH] ), then [EQUATION] Consequently, we have [EQUATION] The penultimate inequality follows from the bound on [MATH] from Lemma 57 and the accuracy of estimates [MATH] This implies that for any [MATH] |
[EQUATION] The second inequality uses that fact that [MATH] is a unit vector whereas [MATH] . Thus, [EQUATION] where [MATH] (from ). Using the fact that [MATH] , we can conclude that |
[EQUATION] Item 2 in the claim is now an immediate consequence. We now give an algorithm which finds out directions [MATH] such that for [MATH] defined before (as [MATH] ), [MATH] is close to a junta on the directions [MATH] |
Lemma 37 The algorithm Find-candidate-directions described in Figure has the following properties: For noise parameter [MATH] , error parameter [MATH] , surface area parameter [MATH] , if the function [MATH] has surface area [MATH] and is a linear [MATH] -junta, then with probability [MATH] , the algorithm outputs vect... |
[MATH] , the function is [MATH] -close to a junta on [MATH] . Further, the directions [MATH] are at least [MATH] linearly independent. The query complexity of this algorithm is [MATH] |
Proof. The bound on the query complexity of this algorithm is immediate by just plugging in the query complexity of the routine test-candidate-direction (Lemma 36 ) and the query complexity of Step 4 (Lemma 10 ). |
Next, observe that by the guarantee of Test-candidate-direction , the set [MATH] output by the algorithm consists of [MATH] -linearly independent directions. Finally, assume that [MATH] is a [MATH] -junta where [MATH] . Then, note that for any [MATH] [MATH] . Now, there are two possibilities: (For the rest of this proo... |
(a) If [MATH] , then note that we have found [MATH] directions [MATH] such that [MATH] . Further, the directions [MATH] are [MATH] -linearly independent. Thus, [MATH] . So, in this case, [MATH] is indeed a junta on [MATH] (where [MATH] ). |
(b) If [MATH] , then we are in one of the two situations: either [MATH] is [MATH] -close to a junta on [MATH] where [MATH] . In this case, we are already done. If not, then we apply Lemma 30 and obtain that with probability at least [MATH] , a randomly chosen direction [MATH] will be at least [MATH] -far from the subsp... |
With the aid of the algorithm Find-candidate-directions , we are able to find implicitly find directions [MATH] such that [MATH] is close to a junta on [MATH] . In the next subsection, we essentially do a hypothesis testing over a set of functions which form a cover for all juntas on [MATH] |
4.1 Hypothesis testing against subspace juntas The following lemma says how given the directions [MATH] and an error parameter [MATH] , we can implicitly find directions which form an orthonormal basis of [MATH] (as before, we are using [MATH] as a shorthand for [MATH] respectively). All the symbols below will have the... |
Lemma 38 Choose any error parameter [MATH] and let [MATH] be [MATH] -linearly independent directions for [MATH] . Then, there is a procedure Compute-ortho-transform which makes [MATH] queries to [MATH] , we can obtain numbers [MATH] such that the following holds: |
1. For [MATH] , all the numbers [MATH] 2. There exists an orthonormal basis [MATH] of [MATH] such that for all [MATH] [EQUATION] |
Proof. Let [MATH] be the function defined in Lemma 57 . Now, observe that [EQUATION] Thus, using Lemma 10 , we can use [MATH] queries to [MATH] to obtain numbers [MATH] such that |
[EQUATION] As [MATH] are [MATH] -linearly independent, hence the vectors [MATH] are [MATH] -linearly independent. With this, we can now apply Lemma 57 to obtain numbers [MATH] such that there is an orthonormal basis [MATH] of [MATH] with the property that (a) |
[MATH] and (b) [MATH] where [MATH] Let us now again set the parameters [MATH] and [MATH] exactly the same as Lemma 37 . Namely, we set [MATH] and [MATH] . With this setting of parameters, we state the following lemma. |
Lemma 39 There is an algorithm Estimate-closest-hypothesis (described in Figure ) which takes as input oracle access to [MATH] , directions [MATH] which are [MATH] -linearly independent, error parameter [MATH] surface area parameter [MATH] . The algorithm has the following guarantee: |
1. It makes [MATH] queries to [MATH] 2. There is an orthonormal basis [MATH] of [MATH] (which is independent of [MATH] ) such that with probability [MATH] , outputs a function [MATH] with the following guarantee: Let [MATH] be the set of functions from Theorem 54 where the parameters [MATH] are set as in Figure . Then, |
[EQUATION] Proof. As usual, the query complexity of the procedure is easily seen to be [MATH] by just plugging in the values of the parameters along with the guarantees on the query complexity of Compute-ortho-transform (Lemma 38 ) as well Lemma 12 |
To analyze the algorithm, let us now define a point [MATH] to be good if the following two conditions hold: 1. For [MATH] [EQUATION] |
2. For all [MATH] [EQUATION] Claim 40 For [MATH] [MATH] Proof. Lemma 12 guarantees that for any specific choice of [MATH] [MATH] . Thus, with probability [MATH] , item [MATH] holds for [MATH] . Likewise, notice that |
[EQUATION] Thus, for any [MATH] , with probability [MATH] , item 2 holds. Thus, by a union bound, it holds for all [MATH] simultaneously, with probability [MATH] . This proves the claim. |
Next, observe that if a point [MATH] is good , then the following holds for every [MATH] [EQUATION] The penultimate inequalities just follow from the condition that [MATH] is good and the values of the parameters. Now, observe that |
[EQUATION] Now, observe that by definition, the term inside the expectation is uniformly bounded by [MATH] . On the other hand, if a point [MATH] is good, then by ( ) and exploiting [MATH] is [MATH] -Lipschitz, then [MATH] . Since the fraction of good points is at least [MATH] , we get that for any [MATH] |
[EQUATION] Now a standard Chernoff bound implies that with for any [MATH] [MATH] is within [MATH] [MATH] with probability [MATH] . Thus, by a union bound, with probability [MATH] , for all [MATH] [MATH] is within [MATH] of [MATH] . This finishes the proof. |
We are now ready to prove Theorem 33 Proof of Theorem 33 Set [MATH] (this is the same setting as Lemma 37 and Lemma 39 ). Observe that with this choice of [MATH] , since [MATH] has surface area bounded by [MATH] , then by Proposition 29 , we get that |
[EQUATION] We now run the algorithm Find-candidate-directions with noise parameter [MATH] , error parameter [MATH] and surface area parameter [MATH] . We are guaranteed that with probability [MATH] , we will get [MATH] directions [MATH] which are [MATH] -linearly independent and [MATH] is [MATH] -close to a junta on th... |
Next, we run the routine Estimate-closest-hypothesis with the directions [MATH] , surface area parameter [MATH] , error parameter [MATH] Observe that the query complexity of Estimate-closest-hypothesis is also [MATH] . Thus, the total query complexity remains [MATH] |
By guarantee of Estimate-closest-hypothesis , we have the following: there is an orthonormal basis [MATH] of [MATH] such that [EQUATION] |
However, conditioned on [MATH] [MATH] is [MATH] -close to a junta on [MATH] . By Theorem 54 , this implies that the quantity [MATH] . This means that if we output the function [MATH] , then |
[MATH] . Consider the subspace [MATH] spanned by vectors [MATH] . Note that [MATH] and [MATH] is a relevant subspace for [MATH] . Thus, [MATH] can be extended to a basis for [MATH] , finishing the proof. |
Remark 41 A crucial point about the routine Find-invariant-structure , which will be useful in the next section, is the following: The marginal distribution of all the queries is distributed as the standard [MATH] -dimensional Gaussian distribution [MATH] . To see this, note that |
1. In the routine Find-candidate-directions , each of the directions [MATH] is sampled from [MATH] . Further, for [MATH] and [MATH] which are i.i.d. samples from [MATH] , the queries made to the oracle for [MATH] in computing [MATH] are also distributed as [MATH] (see Lemma 10 ). |
2. In the routine Estimate-closest-hypothesis , the points [MATH] are sampled from [MATH] as are the directions [MATH] (which are output of Find-candidate-directions ). With this, the queries made to the oracle for [MATH] for computing [MATH] are distributed as [MATH] (see Lemma 12 ). |
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