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{addmargin} [1em]0em – If [MATH] , then, according to GA5, no atom is yielded with [MATH] ; according to the algorithm [MATH] – If [MATH] , then, according to GA5, some atom is yielded with [MATH] according to the algorithm [MATH] |
From these two cases, we have then proved that [MATH] , which is the first condition required for marg to be correct (see C.4 ). |
Let us proof now the second condition, which assumes that [MATH] . The condensation part of the algorithm (lines 2 to 8) entails |
[EQUATION] Using GA4, the [MATH] in the right-hand part can be replaced by probabilities: [EQUATION] Using GA3 (mutual exclusiveness) and the third axiom of probability, we can replace the sum of probabilities by a probability of a disjunction: |
[EQUATION] Merging GA5 and GA6, we establish the following equivalence: [EQUATION] hence [EQUATION] which is the second and last condition required for marg to be correct (see C.4 ). The proof of the correctness of the marg subroutine in proposition C.11 establishes the correctness of the Statues algorithm. |
# Source: arxiv 1806.10057 # Title: Is your function low-dimensional? # Sections: all # Downloaded: 2026-03-03T01:43:56.932489+00:00 |
Is your function low-dimensional? Abstract We study the problem of testing if a function depends on a small number of linear directions of its input data. We call a function [MATH] linear [MATH] -junta if it is completely determined by some [MATH] -dimensional subspace of the input space. In this paper, we study the pr... |
[MATH] - juntas which are functions on the Boolean cube which depend on at most k of the variables and 2. intersection of [MATH] halfspaces, a fundamental geometric concept class. |
We show that the class of linear [MATH] -juntas is not testable, but adding a surface area constraint makes it testable: we give a [MATH] -query non-adaptive tester for linear [MATH] -juntas with surface area at most [MATH] . We show that the polynomial dependence on [MATH] is necessary. Moreover, we show that if the f... |
Introduction Property testing of Boolean functions was initiated in the seminal work of Blum, Luby and Rubinfeld BLR93 and Rubinfeld and Sudan RS96 . The high level goal of property testing is the following: Given (query) access to a Boolean function [MATH] , the algorithm must distinguish between (i) the case that [MA... |
Since the works of BLR93 RS96 , property testing of Boolean functions has been a thriving field and by now several classes [MATH] have been studied from this perspective. These include classes such as linear functions BLR93 , low-degree polynomials JPRZ04 BKS 10 , monotonicity FLN 02 CS16 KMS15 , algebraic properties K... |
Special attention has been devoted to the problem of testing juntas. Recall that a Boolean function [MATH] is said to be a [MATH] -junta if [MATH] is only dependent on a subset [MATH] (of size [MATH] of the coordinates. Given (query) access to a function [MATH] , the problem of testing juntas is to decide whether [MATH... |
[MATH] -juntas can be tested with query complexity independent of [MATH] FKR 04 The most obvious generalization of [MATH] -juntas to functions [MATH] is to consider functions that depend only on [MATH] of the [MATH] coordinates. However, in many statistical and machine learning models (e.g. PCA, ICA, kernel learning, d... |
We note that the family of linear [MATH] -juntas includes important classes of functions that have been studied in the learning and testing literature. Notably it includes: |
Boolean juntas: If [MATH] is a Boolean junta, then [MATH] defined as [MATH] is a linear [MATH] -junta. Functions of halfspaces: Linear [MATH] -juntas include as a special case both halfspaces and intersections of [MATH] -halfspaces. The testability of halfspaces was studied in MORS09 MORS10 RS15 |
We consider the scenario where the ambient dimension [MATH] is large but the dimension of the relevant subspace, i.e., [MATH] is small. In this setting, we consider the following property testing question: |
Question 1 Given a function [MATH] and access to random examples, [MATH] , is it possible to test in number of queries that depends on [MATH] (but not on [MATH] ) if |
[MATH] is a linear [MATH] -junta or far from all linear [MATH] -juntas? The problem of testing linear-juntas is closely related to the problem of model compression in machine learning. The goal of model compression is to take as an input a complex predictor/classifier function and to output a simpler predictor/classifi... |
BCNM06 The question of model compression is extensively studied in the context of deep nets, see e.g., BC14 and follow up work, where the models are often rotationally invariant (with the caveat that the regularization often used in optimization might not be). Thus as a motivating example we may ask: given a complex de... |
To formally state question we need to define what “close” means. The standard definition is to state that [MATH] is close to [MATH] if [MATH] is small, for some probability measure [MATH] . The most natural choice of [MATH] for learning and testing functions |
[MATH] is the Gaussian measure MORS09 KNOW14 Nee14 BBBY12 CFSS17 KOS08 Vem10a DKS18 BL13 HKM12 . It is particularly natural in our setup since the Gaussian measure is invariant under many linear transformation, e.g., all rotations. |
It is possible to show that the answer to question is no even if [MATH] and [MATH] , since without smoothness assumptions, measurable functions [MATH] can look arbitrarily random to any finite number of queries (a more formal statement with stronger results will be discussed shortly). Since the groundbreaking work of K... |
Question 2 Given a function [MATH] and access to random examples, [MATH] , is it possible to test in number of queries that depends on [MATH] and [MATH] (but not on [MATH] ) if [MATH] is close to any linear [MATH] -junta with surface area at most [MATH] |
In our main result we give an affirmative answer to the question above: Theorem 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 distinguishes between ... |
1. The function [MATH] is a linear [MATH] -junta whose surface area is at most [MATH] 2. The function [MATH] is [MATH] -far from any linear [MATH] -junta with surface area at most [MATH] |
This theorem is proven in Section We note that while the tester allows a slack of [MATH] in the surface area between the soundness and completeness cases, such a slack factor is required even for the easier problem of estimating surface area in [MATH] |
Nee14 It is natural to ask if our dependence on the surface area is optimal. Towards answering this, in Section , we prove: Theorem |
Any non-adaptive algorithm for testing whether an unknown Boolean function [MATH] is a linear [MATH] -junta with surface area at most [MATH] versus [MATH] -far from a linear [MATH] -junta makes at least [MATH] queries. |
Thus our tester is optimal in the dependence on [MATH] up to polynomial factors. Finding the linear-invariant structure Given the previous theorem it is natural to ask for more, i.e., not just test if the function is a linear-junta but also find the junta in number of queries that depends only on [MATH] and [MATH] (but... |
[MATH] is close to [MATH] with query complexity independent of [MATH] ? We give an affirmative answer to this question: Theorem Let [MATH] be a linear [MATH] -junta with surface area at most [MATH] . Then, there is an algorithm Find-invariant-structure which on error parameter [MATH] , makes [MATH] queries and outputs ... |
[EQUATION] Moreover, for some [MATH] [EQUATION] Informally, the theorem states that it is possible to find the “linear-invariant” structure (i.e., the structure up to unitary transformation) of [MATH] in number of queries that dependens on [MATH] and [MATH] . Of course, one cannot hope to output the relevant directions... |
Testability of linear invariant families of linear [MATH] -juntas Our ability to find the linear-invariant structure of linear [MATH] -juntas additionally allows us to test subclasses of linear [MATH] -juntas which are closed under rotation. |
Definition 3 Let [MATH] be any collection of functions mapping [MATH] to [MATH] . For any [MATH] let: [EQUATION] Define [MATH] and call it the induced class of [MATH] |
The two key properties of [MATH] are (i) each function [MATH] is a linear [MATH] -junta, (ii) the class [MATH] is closed under unitary transformations. The definition is a continuous analogue of the so-called “induced subclass of [MATH] -dimensional functions” from GOS 09 (that paper was about testing functions over [M... |
Theorem Let [MATH] be a collection of functions mapping [MATH] to [MATH] . Further, for every [MATH] [MATH] . Then, there is an algorithm Test-structure- [MATH] which has the following guarantee: Given oracle access to [MATH] and an error parameter [MATH] , the algorithm makes [MATH] queries and distinguishes between t... |
A particularly important instantiation of the above theorem is the following: Let [MATH] be any collection of functions mapping [MATH] and let [MATH] be defined as |
[EQUATION] Note that [MATH] defined above is the set of functions obtained by composing a function from [MATH] with [MATH] -dimensional halfspaces. Consequently, [MATH] is the of all functions which can be obtained by composing a function from [MATH] with halfspaces. As an example, if [MATH] consists of the [MATH] func... |
Roughly speaking, the algorithm Test-structure- [MATH] works as follows: we first run the routine Test-linear-junta – if the target function [MATH] passes this test, we are guaranteed that it is (very close to) a linear [MATH] -junta with surface area [MATH] . We then run the routine Find-invariant-structure . If the o... |
1.1 Related Work Testing Boolean juntas As we have already mentioned, the problem of testing juntas on [MATH] has already been well-studied. For example, it is known Bla09 CST 17 that [MATH] |
queries are necessary and sufficient for non-adaptively testing [MATH] -juntas with respect to the uniform distribution, while [MATH] queries are necessary and sufficient in the adaptive setting BBM12 It even turns out to be possible to test [MATH] -juntas with respect to an unknown distribution CLS 18 , although in th... |
Learning juntas of half-spaces. There has been extensive work on learning intersections and other functions of [MATH] half-spaces BK97 Vem10b VX13 KOS08 Note that these algorithms (necessarily) require time polynomial in [MATH] (whereas our raison d’etre is a query complexity independent of [MATH] ). In particular, |
BK97 provided conditions under which intersections of halfspaces can be learnt under the uniform distribution on the ball. Vempala Vem10b extended their result to arbitrary log-concave distributions. In terms of the expressivity of the function class, VX13 explicitly considered the problem of learning linear [MATH] -ju... |
[MATH] is learnable in polynomial time if the function [MATH] is identified by low moments and robust to small rotations in [MATH] . Along a related but different axis, KOS08 showed that functions of bounded surface area in the Gaussian space are learnable in polynomial time. Finally, we remark that there also has been... |
Linearly Invariant Testing over Finite Fields We note that the set of linear-juntas is linearly invariant. If [MATH] is a linear [MATH] -junta and [MATH] is any [MATH] matrix then [MATH] is also a linear [MATH] -junta. Over finite fields, KS08 studied general criteria for when a linearly invariant property is testable,... |
BFH 13 . In particular, GOS 09 , gave a [MATH] query complexity algorithm to test linear juntas over finite fields. Moreover, they also show that an exponential lower bound on [MATH] is necessary. This should be contrasted with our result which shows that linear juntas over the Gaussian space can be tested with [MATH] ... |
Testing (functions) of halfspaces The question of testing halfspaces was first considered in MORS10 who showed that in the Gaussian space (as well as the Boolean space), halfspaces are testable with [MATH] queries. Subsequently, the second and third authors (Mossel and Neeman MN15 ) gave a different testing algorithm f... |
1.2 Techniques A major difference between linear juntas over finite fields and linear juntas over Gaussian space is the “infinitesimal geometry” that can be used in the latter and does not exist in the former. In particular, the linear part [MATH] |
of the Hermite expansion of [MATH] is approximately given by [MATH] for large [MATH] . Here [MATH] is the Ornstein-Uhlenbeck operator. Both the quantities, [MATH] and [MATH] can be approximated by sampling a small number of points from the Gaussian distribution and evaluating [MATH] at those points. Moreover, if [MATH]... |
We would like to obtain “many more directions” that lie in the span of [MATH] We do so by considering functions of the form [MATH] , for randomly chosen [MATH] and an appropriate value of [MATH] |
(the experts will recognize [MATH] as part of the definition of the Ornstein-Uhlenbeck operator). Note that [MATH] is also a linear junta defined by the same direction [MATH] and therefore the linear part of the Hermite expansion of [MATH] , is also in the span of [MATH] |
It is now natural to propose the following algorithm to test if a function is a linear [MATH] -junta: choose points [MATH] at random and “compute” |
[MATH] at these points. Then if the rank of the matrix spanned by [MATH] is at most [MATH] , then output YES; otherwise, output NO. |
Of course, actually computing [MATH] requires [MATH] samples. Instead we will approximately compute the Gram matrix [EQUATION] and test if it is close or far from a matrix of rank [MATH] One advantage of using the Gram matrix, is that we can evaluate the entries [MATH] by sampling random inputs to evaluate the expected... |
[EQUATION] How do we know that [MATH] are not very close to [MATH] If [MATH] has a bounded surface area then [MATH] is close to the noise stable function [MATH] . For such noise stable functions, we prove that with good probability at a random point [MATH] |
[MATH] will be of non-negligible size. In fact, one of our main technical lemmas (Lemma 30 ) proves much more. It shows that if [MATH] is [MATH] far from any linear- [MATH] -junta then for any subspace [MATH] with co-dimension at most [MATH] , it holds that for a random [MATH] with probability at least [MATH] , the pro... |
[MATH] . This result is later combined with a perturbation argument to establish to show that if [MATH] is [MATH] -far from a linear [MATH] -junta then indeed the Gram matrix will have [MATH] large eigenvalues. Since our analysis relies on the function [MATH] having surface area at most [MATH] , the first stage of the ... |
The algorithm to identify the linear invariant structure of [MATH] builds up on the ideas in the algorithm to test linear [MATH] -juntas. More precisely, we can show that if [MATH] is a linear [MATH] -junta with surface area [MATH] |
1. we can find directions [MATH] such that [MATH] is close to a function on the space spanned by the directions [MATH] (for some [MATH] ). |
2. While we cannot find [MATH] explicitly for any [MATH] , we can evaluate [MATH] at any point [MATH] up to good accuracy. 3. With the above observation, the high level idea is to try out all smooth functions on the subspace spanned by [MATH] . Perform hypothesis testing for each such function against [MATH] and output... |
The crucial part in the above argument is that even if we have [MATH] implicitly, the space of “all smooth functions” on [MATH] has a cover whose size is independent of [MATH] . This lets us identify the linear invariant function defining [MATH] with query complexity just dependent on [MATH] and [MATH] |
In order to prove lower bounds in terms of surface area, we construct a distribution over linear [MATH] -juntas with large surface area by splitting [MATH] into many very thin parallel strips (oriented in a random direction) and assign our function a random [MATH] value on each strip. (Note that the surface area of suc... |
Preliminaries In this paper, unless explicitly mentioned otherwise, the domain [MATH] is always endowed with the measure [MATH] , the standard [MATH] -dimensional Gaussian measure. Likewise, we will only consider functions [MATH] . For such a function, and [MATH] , we recall that the so-called Ornstein-Uhlenbeck operat... |
[EQUATION] We will also need to recall some very basic facts about Hermite expansion for functions [MATH] . In particular, recall that for all [MATH] , we can define the Hermite polynomial [MATH] as |
[EQUATION] Further, for the ambient space [MATH] , let us define the space [MATH] to be the linear subspace of [MATH] spanned by [MATH] . Here [MATH] denotes the unit sphere in [MATH] -dimensions. For a function [MATH] , we let [MATH] denote the projection of [MATH] to the subspace [MATH] . Note that for any [MATH] [MA... |
Proposition 4 1. For [MATH] , the subspaces [MATH] and [MATH] are orthogonal. In other words, if [MATH] and [MATH] , then [MATH] |
2. Every function [MATH] can be expressed as [MATH] where [MATH] is the projection of [MATH] to [MATH] 3. For any [MATH] [MATH] 2.0.1 Oracle computation |
We now list several useful claims which all fit the same motif: Given oracle access to [MATH] , what interesting quantities can be computed? |
Lemma 5 Given oracle access to [MATH] , error parameter [MATH] , there is a function [MATH] such that the following holds for every [MATH] |
[EQUATION] Further, for any [MATH] , we can compute [MATH] to additive error [MATH] with confidence [MATH] by making [MATH] queries to the oracle for [MATH] |
Proof. Observe that for any [MATH] [MATH] . This implies that [EQUATION] Set [MATH] so that [MATH] and let us define [MATH] as [MATH] Now, observe that for [MATH] [MATH] and [MATH] . We now apply Chebyshev’s inequality to obtain |
[EQUATION] Next, observe that both [MATH] and [MATH] can be computed to error [MATH] with confidence [MATH] using [MATH] queries to the oracle for [MATH] . This immediately implies that [MATH] can be computed to error [MATH] using [MATH] queries to the oracle for [MATH] |
Lemma 6 Given oracle access to functions [MATH] , error parameter [MATH] and confidence parameter [MATH] , there is an algorithm which makes [MATH] queries to [MATH] and computes [MATH] up to error [MATH] with confidence [MATH] |
Proof. Consider the function [EQUATION] Writing out the Fourier expansions of [MATH] and [MATH] , note that [MATH] and so [EQUATION] |
Since [MATH] and [MATH] are linear functions, [MATH] . On the other hand, [MATH] , and so the Cauchy-Schwarz inequality implies that |
[EQUATION] Hence, [MATH] . If we choose [MATH] so that [MATH] , then it only remains to show that we can estimate [MATH] within additive error [MATH] with confidence [MATH] |
Let [MATH] and [MATH] be Gaussian random variables, independent of [MATH] , and write [MATH] and [MATH] . In particular, we can express [MATH] in the form [MATH] where |
[EQUATION] Recalling that [MATH] , it follows that [MATH] takes values in [MATH] and it follows from Hoeffding’s inequality that we can approximate [MATH] to additive error [MATH] |
with confidence [MATH] using [MATH] samples of [MATH] . Moreover, each sample of [MATH] can be computed using two oracle queries to [MATH] and two oracle queries to [MATH] |
Definition 7 A function [MATH] is said to be a linear [MATH] -junta if there are at most [MATH] orthonormal vectors [MATH] and a function [MATH] such that |
[EQUATION] Further, if [MATH] (a linear subspace of [MATH] ), then [MATH] is said to be a [MATH] -junta. 2.1 Derivatives of functions |
We will use [MATH] to denote the derivative operator. In case, there are two sets of variables involved, we will explicitly indicate the variable with respect to which we are taking the derivative. |
Definition 8 For [MATH] [MATH] ) and [MATH] , define the function [MATH] [EQUATION] Further, in the same setting as above, we let |
[MATH] [EQUATION] Let [MATH] denote the derivative operator with respect to [MATH] and let [MATH] denote the derivative operator with respect to [MATH] . Then, it is easy to observe that |
[EQUATION] Next, for a function [MATH] , define [MATH] as the degree- [MATH] Hermite coefficients of [MATH] . In other words, the [MATH] coordinate of [MATH] |
[EQUATION] where [MATH] , the standard [MATH] -dimensional Gaussian measure. With respect to our earlier definition of [MATH] , observe that we have: |
[MATH] We next prove the following important lemma which connects the gradient of [MATH] at [MATH] with [MATH] . In particular, we have the following lemma. |
Lemma 9 [EQUATION] Proof. First of all, observe that for any function [MATH] with bounded derivatives, and for any [MATH] [EQUATION] |
While the first and last equalities are trivial, the middle is a consequence of integration by parts. Assuming that [MATH] has bounded derivatives, we may apply this identity to [MATH] , yielding |
[EQUATION] This proves the lemma in the case that [MATH] has bounded derivatives. In the general case, we approximate choose a sequence of functions that have bounded derivatives and approximate [MATH] in [MATH] Applying the lemma to these functions and taking the limit proves the general case. |
Lemma 10 Given oracle access to [MATH] , noise parameter [MATH] , error parameter [MATH] , confidence parameter [MATH] and [MATH] , there is an algorithm which makes [MATH] queries to [MATH] and computes [MATH] up to error [MATH] with confidence [MATH] |
Proof. By Lemma , we have [EQUATION] We can now apply Lemma to finish the proof. Proposition 11 For any [MATH] [MATH] Proof. By Lemma , we have [MATH] . Now, observe that the range of [MATH] is [MATH] and thus, [MATH] , implying the stated upper bound. |
Lemma 12 Given oracle access to [MATH] [MATH] , noise parameter [MATH] error parameter [MATH] , there is a function [MATH] such that the following holds for every [MATH] |
[EQUATION] Further, for an error parameter [MATH] , confidence parameter [MATH] , we can compute [MATH] to additive error [MATH] with confidence [MATH] using [MATH] queries to [MATH] |
Proof. We first use Lemma and obtain that [EQUATION] Consequently, we have that [EQUATION] The claim now follows from Lemma 2.2 Some useful inequalities concerning noise stability |
Lemma 13 [Poincaré inequality] Let [MATH] be a [MATH] function. Then, [MATH] Definition 14 For a Borel set [MATH] , we define its Gaussian surface area [MATH] to be |
[EQUATION] provided the limit exists. Here, for any body [MATH] [MATH] denotes the Gaussian volume of [MATH] , i.e., [MATH] . Further, [MATH] where [MATH] denotes the Euclidean distance of [MATH] from [MATH] |
For a function [MATH] , we denote its surface area [MATH] where [MATH] Ledoux Led94 (and implicitly Pisier Pis86 ) proved the following connection between noise sensitivity and surface area of functions. |
Lemma 15 [Ledoux Led94 For any [MATH] and [MATH] [MATH] , we have [EQUATION] The following proposition is an immediate consequence of the above lemma. |
Proposition 16 Let [MATH] [MATH] and [MATH] . Then, 1. [MATH] 2. For any [MATH] and [MATH] [MATH] Proof. Let [MATH] denote the event that |
[MATH] To prove the first item, observe that for any [MATH] [EQUATION] Thus, we obtain that [EQUATION] where the last inequality is an application of Lemma 15 . The second item here is the same as Theorem 15 (full version) of KOS08 . So, we do not prove it here. |
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