Original: List-Decodable Sparse Mean Estimation
Revised: List-Decodable Sparse Mean Estimation

Original: Anonymous Author(s)
Revised: Anonymous Author(s)

Original: AfﬁliationAddress email
Revised: AfﬁliationAddress email

Original: Abstract
Revised: Abstract

Original: Robust mean estimation is one of the most important problems in statistics: givena set of samples { x 1 , . . .
Revised: Robust mean estimation is one of the most important problems in statistics: givena set of samples { x 1 , . . .

Original: , x n } ⇢ R d where an ↵ fraction are drawn from somedistribution D and the rest are adversarially corrupted, it aims to estimate themean of D .
Revised: , x n } ⇢ R d where an ↵ fraction are drawn from somedistribution D and the rest are adversarially corrupted, it aims to estimate themean of D .

Original: A surge of recent research interest has been focusing on the listdecodable setting where ↵ 2 (0 , 12 ] , and the goal is to output a ﬁnite number ofestimates among which at least one approximates the target mean.
Revised: A surge of recent research interest has been focusing on the listdecodable setting where ↵ 2 (0 , 12 ] , and the goal is to output a ﬁnite number ofestimates among which at least one approximates the target mean.

Original: In this paper,we consider that the underlying distribution is Gaussian and the target mean isk -sparse.
Revised: In this paper,we consider that the underlying distribution is Gaussian and the target mean isk -sparse.

Original: Our main contribution is the ﬁrst polynomial-time algorithm that enjoyssample complexity O
Revised: Our main contribution is the ﬁrst polynomial-time algorithm that enjoyssample complexity O

Original:  poly( k, log d ) 
Revised:  poly( k, log d )

Original: , i.e. poly-logarithmic in the dimension.
Revised: , i.e. poly-logarithmic in the dimension.

Original: Oneof the main algorithmic ingredients is using low-degree sparse polynomials to ﬁlteroutliers, which may be of independent interest.
Revised: Oneof the main algorithmic ingredients is using low-degree sparse polynomials to ﬁlteroutliers, which may be of independent interest.

Original:  Introduction
Revised:  Introduction

Original: Mean estimation is arguably one of the most fundamental inference tasks in statistics and machinelearning.
Revised: Mean estimation is arguably one of the most fundamental inference tasks in statistics and machinelearning.

Original: Given a set of samples { x 1 , . . .
Revised: Given a set of samples { x 1 , . . .

Original: , x n } ⇢ R d where an ↵ fraction are drawn from somewell-behaved (e.g. Gaussian) distribution D and the rest are adversarially corrupted, the goal is toestimate the mean of D .
Revised: , x n } ⇢ R d where an ↵ fraction are drawn from somewell-behaved (e.g. Gaussian) distribution D and the rest are adversarially corrupted, the goal is toestimate the mean of D .

Original: In the noiseless case where ↵ = 1 , the problem can be easily solved in viewof the concentration of measure phenomenon [LT91].
Revised: In the noiseless case where ↵ = 1 , the problem can be easily solved in viewof the concentration of measure phenomenon [LT91].

Original: However, this is rarely the case in practicebecause modern data sets are often contaminated by random noise or even by adversarial corruptions.
Revised: However, this is rarely the case in practicebecause modern data sets are often contaminated by random noise or even by adversarial corruptions.

Original: Thus, a great deal of recent efforts are focused on efﬁciently and robustly estimating the target meanin the presence of outliers.
Revised: Thus, a great deal of recent efforts are focused on efﬁciently and robustly estimating the target meanin the presence of outliers.

Original: Generally speaking, there is a phase transition between ↵ > 1 / 2 and 0 < ↵  1 / 2 , and solving eitherproblem in a computationally efﬁcient manner is highly nontrivial.
Revised: Generally speaking, there is a phase transition between ↵ > 1 / 2 and 0 < ↵  1 / 2 , and solving eitherproblem in a computationally efﬁcient manner is highly nontrivial.

Original: The problem that most of thesamples are uncorrupted, i.e. ↵ > 1 / 2 , has a very long history dating back to 1960s [Tuk60, Hub64],yet only until recently have computationally efﬁcient algorithms been established [DKK + 16, LRV16].
Revised: The problem that most of thesamples are uncorrupted, i.e. ↵ > 1 / 2 , has a very long history dating back to 1960s [Tuk60, Hub64],yet only until recently have computationally efﬁcient algorithms been established [DKK + 16, LRV16].

Original: The other yet more challenging regime concerns that an overwhelming fraction of the samples arecorrupted, i.e. ↵  1 / 2 , which even renders learning impossible.
Revised: The other yet more challenging regime concerns that an overwhelming fraction of the samples arecorrupted, i.e. ↵  1 / 2 , which even renders learning impossible.

Original: This motivates the researchon list-decodable mean estimation [CSV17], where in place of outputting one single estimate, thealgorithm is allowed to generate a list of candidates and is considered to be successful if there existsat least one candidate in the list that is sufﬁciently close to the true mean.
Revised: This motivates the researchon list-decodable mean estimation [CSV17], where in place of outputting one single estimate, thealgorithm is allowed to generate a list of candidates and is considered to be successful if there existsat least one candidate in the list that is sufﬁciently close to the true mean.

Original: In this work, we investigate the problem of list-decodable mean estimation, for which there hasbeen a plethora of elegant results established in recent years.
Revised: In this work, we investigate the problem of list-decodable mean estimation, for which there hasbeen a plethora of elegant results established in recent years.

Original: From a high level, most of themconcern error guarantees and running time.
Revised: From a high level, most of themconcern error guarantees and running time.

Original: For example, [CSV17] proposed the ﬁrst tractablealgorithm based on the ellipsoid method, which runs in polynomial time and achieves optimal errorrate for variance-bounded distributions.
Revised: For example, [CSV17] proposed the ﬁrst tractablealgorithm based on the ellipsoid method, which runs in polynomial time and achieves optimal errorrate for variance-bounded distributions.

Original: [DKS18b] developed a multi-ﬁltering scheme and showedthat the error rate can be improved by using higher-order degree polynomials if the underlyingdistribution is Gaussian.
Revised: [DKS18b] developed a multi-ﬁltering scheme and showedthat the error rate can be improved by using higher-order degree polynomials if the underlyingdistribution is Gaussian.

Original: The more recent works [CMY20, DKK20a, DKK + 21a] further addressedthe computational efﬁciency of this task and achieved almost linear running time.
Revised: The more recent works [CMY20, DKK20a, DKK + 21a] further addressedthe computational efﬁciency of this task and achieved almost linear running time.

Original: Although all of these algorithms exhibit near-optimal guarantees on either error rate or computationalcomplexity, it seems that less is explored to improve another yet important metric: the samplecomplexity.
Revised: Although all of these algorithms exhibit near-optimal guarantees on either error rate or computationalcomplexity, it seems that less is explored to improve another yet important metric: the samplecomplexity.

Original: In particular, the sample complexity of all these algorithms is O (poly( d )) , hence theyquickly break down for data-demanding applications such as healthcare where the number of availablesamples n is typically orders of magnitude less than the dimension d
Revised: In particular, the sample complexity of all these algorithms is O (poly( d )) , hence theyquickly break down for data-demanding applications such as healthcare where the number of availablesamples n is typically orders of magnitude less than the dimension d

Original: [Wai19].
Revised: [Wai19].

Original: Therefore, a pressingquestion that needs to be addressed in such high-dimensional regime 1 is the following:
Revised: Therefore, a pressingquestion that needs to be addressed in such high-dimensional regime 1 is the following:

Original: Does there exist a provably robust algorithm for list-decodable mean estimation thatruns in polynomial time and enjoys a sample complexity bound of O (polylog( d )) ?
Revised: Does there exist a provably robust algorithm for list-decodable mean estimation thatruns in polynomial time and enjoys a sample complexity bound of O (polylog( d )) ?

Original: In this paper, we answer the question in the afﬁrmative by showing that when the true mean µ isk -sparse, i.e. it has at most k non-zero elements, it is attribute-efﬁciently list-decodable.
Revised: In this paper, we answer the question in the afﬁrmative by showing that when the true mean µ isk -sparse, i.e. it has at most k non-zero elements, it is attribute-efﬁciently list-decodable.

Original: Theorem 1 (Main result) .
Revised: Theorem 1 (Main result) .

Original: Given the parameter ↵ 2 (0 , 12 ] , ` 2 N , failure probability ⌧ , and a set
Revised: Given the parameter ↵ 2 (0 , 12 ] , ` 2 N , failure probability ⌧ , and a set

Original: T of ⌦
Revised: T of ⌦

Original: [Equation]
Revised: [Equation]

Original: ⌘ samples in R d , of which at least a (2 ↵ ) -fractionare independent draws from the standard Gaussian N ( µ, I d ) , where k µ k 0  k , there exists an 50 algorithm that runs in time poly ( | T | , d ) and returns a list of O (1 / ↵ ) candidate vectors such that,with probability at least 1  ⌧ , the list contains at least one ˆ µ 2 R d with k ˆ µ  µ k 

Revised: ⌘ samples in R d , of which at least a (2 ↵ ) -fractionare independent draws from the standard Gaussian N ( µ, I d ) , where k µ k 0  k , there exists an 50 algorithm that runs in time poly ( | T | , d ) and returns a list of O (1 / ↵ ) candidate vectors such that,with probability at least 1  ⌧ , the list contains at least one ˆ µ 2 R d with k ˆ µ  µ k

Original: = ˜
Revised: = ˜

Original: O `  ↵  12 `  .
Revised: O `  ↵  12 `  .

Original: Remark 2.
Revised: Remark 2.

Original: The key message of the theorem is that when the true mean is k -sparse, it is possible toefﬁciently approximate it with O (poly( k, log d )) samples.
Revised: The key message of the theorem is that when the true mean is k -sparse, it is possible toefﬁciently approximate it with O (poly( k, log d )) samples.

Original: This is in stark contrast to existing listdecodable results [CSV17, DKS18b, CMY20, DKK20a, DKK + 21a] where the sample complexityis
Revised: This is in stark contrast to existing listdecodable results [CSV17, DKS18b, CMY20, DKK20a, DKK + 21a] where the sample complexityis

Original: O (poly( d )) – it is unclear how these algorithms can be made attribute-efﬁcient.
Revised: O (poly( d )) – it is unclear how these algorithms can be made attribute-efﬁcient.

Original: The only attributeefﬁcient robust mean estimators are [BDLS17, DKK + 19, CDK + 21], but their results only hold forthe mild corruption regime where ↵ > 1 / 2 .
Revised: The only attributeefﬁcient robust mean estimators are [BDLS17, DKK + 19, CDK + 21], but their results only hold forthe mild corruption regime where ↵ > 1 / 2 .

Original: Remark 3.
Revised: Remark 3.

Original: As opposed to an O ( ✏ ) error rate obtained for ↵ > 1 / 2 , the (non-vanishing) error rate˜ O `  ↵  12 `  is typically what one can expect for list-decodable mean estimation under bounded (2 ` ) -thorder moment condition, in light of the lower bounds established in [DKS18b].
Revised: As opposed to an O ( ✏ ) error rate obtained for ↵ > 1 / 2 , the (non-vanishing) error rate˜ O `  ↵  12 `  is typically what one can expect for list-decodable mean estimation under bounded (2 ` ) -thorder moment condition, in light of the lower bounds established in [DKS18b].

Original: 1.1 Overview of Techniques
Revised: 1.1 Overview of Techniques

Original: The existing techniques for robust mean estimation can be divided into two categories: one is toformulate and solve the problem by convex optimization [DKK + 16, CSV17, CMY20, CDK + 21];the other is iteratively ﬁltering the outliers until the samples are sufﬁciently concentrated [DKK + 16,
Revised: The existing techniques for robust mean estimation can be divided into two categories: one is toformulate and solve the problem by convex optimization [DKK + 16, CSV17, CMY20, CDK + 21];the other is iteratively ﬁltering the outliers until the samples are sufﬁciently concentrated [DKK + 16,

Original: LRV16, DKS18b, DKK + 19, DKK20a, DKK + 21a].
Revised: LRV16, DKS18b, DKK + 19, DKK20a, DKK + 21a].

Original: Our techniques fall into the latter.
Revised: Our techniques fall into the latter.

Original: Amongall these prior works, the closest to ours are [DKS18b, DKK + 19], which use iterative ﬁlteringframework and leverage low-degree polynomials to ﬁlter outliers.
Revised: Amongall these prior works, the closest to ours are [DKS18b, DKK + 19], which use iterative ﬁlteringframework and leverage low-degree polynomials to ﬁlter outliers.

Original: However, [DKS18b] requires asample complexity polynomial in the sample dimension d , and [DKK + 19] only works under themild corruption regime.
Revised: However, [DKS18b] requires asample complexity polynomial in the sample dimension d , and [DKK + 19] only works under themild corruption regime.

Original: In this work, we propose an attribute-efﬁcient algorithm which addressesoverwhelming corruptions.
Revised: In this work, we propose an attribute-efﬁcient algorithm which addressesoverwhelming corruptions.

Original: Sparse polynomials.
Revised: Sparse polynomials.

Original: To ensure that our algorithm is attribute-efﬁcient, all the polynomials we useto ﬁlter are sparse, i.e. consisting of at most poly( k ) monomials.
Revised: To ensure that our algorithm is attribute-efﬁcient, all the polynomials we useto ﬁlter are sparse, i.e. consisting of at most poly( k ) monomials.

Original: To do this, we will measurethe discrepancy between the empirical covariance matrix of our sample set and the Gaussian, andidentify a subset of coordinates where the discrepancies are the most signiﬁcant.
Revised: To do this, we will measurethe discrepancy between the empirical covariance matrix of our sample set and the Gaussian, andidentify a subset of coordinates where the discrepancies are the most signiﬁcant.

Original: We then constructa polynomial of which the monomials only include coordinates from the subset.
Revised: We then constructa polynomial of which the monomials only include coordinates from the subset.

Original: This would besufﬁcient for us to remove the outliers until the samples become close to Gaussian.
Revised: This would besufﬁcient for us to remove the outliers until the samples become close to Gaussian.

Original: Moreover, wedevelop a set of new results which bound the function values of our sparse polynomials over an
Revised: Moreover, wedevelop a set of new results which bound the function values of our sparse polynomials over an

Original: L 1 -distance bounded sample set, given that the function value is well concentrated on some Gaussiandistribution.
Revised: L 1 -distance bounded sample set, given that the function value is well concentrated on some Gaussiandistribution.

Original: This proves to be critical for distinguishing between good samples and outliers.
Revised: This proves to be critical for distinguishing between good samples and outliers.

Original: Quadratic multiﬁlter with Hermite basis.
Revised: Quadratic multiﬁlter with Hermite basis.

Original: When ﬁltering on a subset of coordinates, ﬁnding theexact set of coordinates, i.e. ⌦ ✓ [ d ] , [ d ] := { 1 , . . .
Revised: When ﬁltering on a subset of coordinates, ﬁnding theexact set of coordinates, i.e. ⌦ ✓ [ d ] , [ d ] := { 1 , . . .

Original: , d } , where the samples behave mostly unlikea Gaussian is an NP-hard problem.
Revised: , d } , where the samples behave mostly unlikea Gaussian is an NP-hard problem.

Original: To rectify this issue, instead of directly searching for ⌦ , wewill embed the samples in a higher-dimensional space and search for an index set
Revised: To rectify this issue, instead of directly searching for ⌦ , wewill embed the samples in a higher-dimensional space and search for an index set

Original: U ✓ [ d ] ⇥ [ d ] .
Revised: U ✓ [ d ] ⇥ [ d ] .

Original: Speciﬁcally, we will subtract the identity matrix I d from the empirical covariance matrix and retrievethe largest entries to construct a quadratic polynomial , which should be more sensitive to a smallamount of outliers than the linear projections [DKS18b, DKK + 19].
Revised: Speciﬁcally, we will subtract the identity matrix I d from the empirical covariance matrix and retrievethe largest entries to construct a quadratic polynomial , which should be more sensitive to a smallamount of outliers than the linear projections [DKS18b, DKK + 19].

Original: This would sufﬁce for removingenough amount of outliers without paying the price of an exhaustive search.
Revised: This would sufﬁce for removingenough amount of outliers without paying the price of an exhaustive search.

Original: The key challenge here really lies in how to estimate the variance of a quadratic polynomial on the
Revised: The key challenge here really lies in how to estimate the variance of a quadratic polynomial on the

Original: Gaussian samples, as it depends on the unknown mean (which is actually what we are estimating).
Revised: Gaussian samples, as it depends on the unknown mean (which is actually what we are estimating).

Original: In other words, if the empirical mean lies far away from the true mean, the good samples can bescattered by applying the quadratic polynomial to them.
Revised: In other words, if the empirical mean lies far away from the true mean, the good samples can bescattered by applying the quadratic polynomial to them.

Original: This is problematic as the algorithm mayﬁlter out a large amount of good samples while keeping most of the outliers.
Revised: This is problematic as the algorithm mayﬁlter out a large amount of good samples while keeping most of the outliers.

Original: Hence, we need tocarefully select appropriate polynomials to ﬁlter.
Revised: Hence, we need tocarefully select appropriate polynomials to ﬁlter.

Original: That is, we require the quadratic polynomials to beharmonic (consisting of only Hermite basis) in light of an observation in that estimating the varianceof a harmonic polynomial is equivalent to estimating the expectation of several relevant multilinearpolynomials [DKS18b].
Revised: That is, we require the quadratic polynomials to beharmonic (consisting of only Hermite basis) in light of an observation in that estimating the varianceof a harmonic polynomial is equivalent to estimating the expectation of several relevant multilinearpolynomials [DKS18b].

Original: Pruning by L 1 -norm.
Revised: Pruning by L 1 -norm.

Original: To bound the function value of a sparse polynomial, we leverage a keyproperty of the Gaussian samples that they are contained in a L 1 -ball with high probability.
Revised: To bound the function value of a sparse polynomial, we leverage a keyproperty of the Gaussian samples that they are contained in a L 1 -ball with high probability.

Original: We thusdeﬁne a notion of ↵ -good set if at least an ↵ -fraction of it are the good samples and the L 1 -distancebetween the samples is bounded.
Revised: We thusdeﬁne a notion of ↵ -good set if at least an ↵ -fraction of it are the good samples and the L 1 -distancebetween the samples is bounded.

Original: Notice that we do not require the input set to be spacially bounded- it only has to include a noticeable ↵ fraction that are drawn from N ( µ, I d ) .
Revised: Notice that we do not require the input set to be spacially bounded- it only has to include a noticeable ↵ fraction that are drawn from N ( µ, I d ) .

Original: We then introduce aclustering step that measures the L 1 -distance between the samples and is guaranteed to generatean initial list of subsets among which at least one is ↵ -good.
Revised: We then introduce aclustering step that measures the L 1 -distance between the samples and is guaranteed to generatean initial list of subsets among which at least one is ↵ -good.

Original: This would sufﬁce for bounding thefunction value of any sparse polynomial.
Revised: This would sufﬁce for bounding thefunction value of any sparse polynomial.

Original: Reader can refer to Section 3.3 for more details.
Revised: Reader can refer to Section 3.3 for more details.

Original:  Related Works
Revised:  Related Works

Original: Breaking the barrier of the typical O (poly( d )) sample complexity bound is one of the centralproblems across many ﬁelds of science and engineering.
Revised: Breaking the barrier of the typical O (poly( d )) sample complexity bound is one of the centralproblems across many ﬁelds of science and engineering.

Original: Motivated by real-world applications,a property termed sparsity is often assumed for this end, meaning that only k out of the dnumber of attributes contribute to the underlying inference problem.
Revised: Motivated by real-world applications,a property termed sparsity is often assumed for this end, meaning that only k out of the dnumber of attributes contribute to the underlying inference problem.

Original: In this way, an improvedbound of O (poly( k, log d )) can be obtained in many inference paradigms such as linear regression [CDS98, Tib96, CT05, Don06, PV13, PVY17], learning of threshold functions [Lit87, BHL95,
Revised: In this way, an improvedbound of O (poly( k, log d )) can be obtained in many inference paradigms such as linear regression [CDS98, Tib96, CT05, Don06, PV13, PVY17], learning of threshold functions [Lit87, BHL95,

Original: STT12, ABHZ16, ZSA20, SZ21], principal component analysis [Ma13, DKK + 19], and mean estimation [BDLS17, DKK + 19, CDK + 21].
Revised: STT12, ABHZ16, ZSA20, SZ21], principal component analysis [Ma13, DKK + 19], and mean estimation [BDLS17, DKK + 19, CDK + 21].

Original: Unfortunately, the success of all these attribute-efﬁcientalgorithms hinges on the presumption that the majority of the data are clean, hence they immediatelyfail to provide any performance guarantee in the list-decodable setting.
Revised: Unfortunately, the success of all these attribute-efﬁcientalgorithms hinges on the presumption that the majority of the data are clean, hence they immediatelyfail to provide any performance guarantee in the list-decodable setting.

Original: Learning with mild corruption ( ↵ > 1 / 2 ).
Revised: Learning with mild corruption ( ↵ > 1 / 2 ).

Original: Learning with noise has been extensively studied in abroad context.
Revised: Learning with noise has been extensively studied in abroad context.

Original: In supervised learning where a sample consists of an instance (i.e. feature vector)and a label, lots of research efforts were dedicated to robust algorithms under label noise, such asrandom classiﬁcation noise [AL87, BFKV96] and Massart noise [Slo88, MN06], where each timethe adversary must ﬂip a coin to see if it is allowed to ﬂip the label.
Revised: In supervised learning where a sample consists of an instance (i.e. feature vector)and a label, lots of research efforts were dedicated to robust algorithms under label noise, such asrandom classiﬁcation noise [AL87, BFKV96] and Massart noise [Slo88, MN06], where each timethe adversary must ﬂip a coin to see if it is allowed to ﬂip the label.

Original: Recent years have witnessedsigniﬁcant progress towards optimal algorithms in the presence of label noise, see for example,[KKMS05, ABL17, DKTZ20, ZSA20, DKK + 20b] and the many references therein.
Revised: Recent years have witnessedsigniﬁcant progress towards optimal algorithms in the presence of label noise, see for example,[KKMS05, ABL17, DKTZ20, ZSA20, DKK + 20b] and the many references therein.

Original: The regime that both instances and labels are corrupted turns out to be signiﬁcantly more challenging.
Revised: The regime that both instances and labels are corrupted turns out to be signiﬁcantly more challenging.

Original: The problem of learning halfspaces under such setting was put forward in 1990s [Val85, KV88],yet only until recently have efﬁcient algorithms been established with dimension-independent noisetolerance [ABL17, DKS18a, She21].
Revised: The problem of learning halfspaces under such setting was put forward in 1990s [Val85, KV88],yet only until recently have efﬁcient algorithms been established with dimension-independent noisetolerance [ABL17, DKS18a, She21].

Original: In addition, [BJK15, KKM18, LSLC20] studied robust linearregression and [BDLS17] presented a set of interesting results regarding robustness under variousstatistical models such as generalized linear models.
Revised: In addition, [BJK15, KKM18, LSLC20] studied robust linearregression and [BDLS17] presented a set of interesting results regarding robustness under variousstatistical models such as generalized linear models.

Original: More in line with this work is robust meanestimation, see the breakthrough works of [DKK + 16, LRV16] and many follow-up works [DKK + 17,
Revised: More in line with this work is robust meanestimation, see the breakthrough works of [DKK + 16, LRV16] and many follow-up works [DKK + 17,

Original: BDLS17, DKS17, SCV18, KSS18, DKK + 19, HLZ20, CDK + 21].
Revised: BDLS17, DKS17, SCV18, KSS18, DKK + 19, HLZ20, CDK + 21].

Original: Learning with overwhelming corruption ( ↵  1 / 2 ).
Revised: Learning with overwhelming corruption ( ↵  1 / 2 ).

Original: The agnostic label noise of [Hau92, KSS92]seems the earliest notion that allows the adversary to arbitrarily corrupt any fraction of the data (say70% ), though it can only corrupt labels.
Revised: The agnostic label noise of [Hau92, KSS92]seems the earliest notion that allows the adversary to arbitrarily corrupt any fraction of the data (say70% ), though it can only corrupt labels.

Original: Following [CSV17], a lot of recent works have studiedthe scenario that both instances and labels are grossly corrupted, and the goal is to output a list ofcandidate parameters among which at least one is a good approximation to the target.
Revised: Following [CSV17], a lot of recent works have studiedthe scenario that both instances and labels are grossly corrupted, and the goal is to output a list ofcandidate parameters among which at least one is a good approximation to the target.

Original: This includeslist-decodable learning of mixture models [DKS18b, DKK + 21b], regression [KKK19, RY20a], andsubspace recovery [RY20b, BK21].
Revised: This includeslist-decodable learning of mixture models [DKS18b, DKK + 21b], regression [KKK19, RY20a], andsubspace recovery [RY20b, BK21].

Original: [DKP + 21] also established a statistical query low bound forlist-decodable linear regression that qualitatively matches the performance of existing algorithms.
Revised: [DKP + 21] also established a statistical query low bound forlist-decodable linear regression that qualitatively matches the performance of existing algorithms.

Original: Interestingly, there are some works studying the problem under the crowdsourced learning settings,where the samples are collected from crowd workers and most of them behave adversarially [SVC16,
Revised: Interestingly, there are some works studying the problem under the crowdsourced learning settings,where the samples are collected from crowd workers and most of them behave adversarially [SVC16,

Original: MV18, ZS21].
Revised: MV18, ZS21].

Original:  Preliminaries
Revised:  Preliminaries

Original: For a vector v , denote by k v k 2 its L 2 -norm, k v k 1 its L 1 -norm, k v k 

Revised: For a vector v , denote by k v k 2 its L 2 -norm, k v k 1 its L 1 -norm, k v k

Original: its zero “norm” that counts the 144 number of non-zeros, and k v k 1 its inﬁnity norm.
Revised: its zero “norm” that counts the 144 number of non-zeros, and k v k 1 its inﬁnity norm.

Original: For a matrix M , denote by k M k 2 its spectral 145 norm, k M k F its Frobenius norm and k M k ⇤ its nuclear norm.
Revised: For a matrix M , denote by k M k 2 its spectral 145 norm, k M k F its Frobenius norm and k M k ⇤ its nuclear norm.

Original: The hard thresholding operatorhard k : R d !
Revised: The hard thresholding operatorhard k : R d !

Original: R d keeps the k largest elements (in magnitude) of a vector and sets the remaining tozero.
Revised: R d keeps the k largest elements (in magnitude) of a vector and sets the remaining tozero.

Original: Deﬁne [ d ] := { 1 , 2 , . . .
Revised: Deﬁne [ d ] := { 1 , 2 , . . .

Original: , d } .
Revised: , d } .

Original: For index set ⌦ ✓ [ d ] , v ⌦ can be regarded as an | ⌦ | -dimensionalvector or a d -dimensional vector with non-zeros on ⌦ .
Revised: For index set ⌦ ✓ [ d ] , v ⌦ can be regarded as an | ⌦ | -dimensionalvector or a d -dimensional vector with non-zeros on ⌦ .

Original: For U ✓ [ d ] ⇥ [ d ] , denote by M U the submatrixof M with entries restricted to U .
Revised: For U ✓ [ d ] ⇥ [ d ] , denote by M U the submatrixof M with entries restricted to U .

Original: We reserve the capital letter G for a random draw of N ( µ, I d ) , i.e. G ⇠ N ( µ, I d ) , where µ 

Revised: We reserve the capital letter G for a random draw of N ( µ, I d ) , i.e. G ⇠ N ( µ, I d ) , where µ

Original: R d , k µ k 0  k is reserved for the target mean that we aim to estimate.
Revised: R d , k µ k 0  k is reserved for the target mean that we aim to estimate.

Original: 3.1 Polynomials
Revised: 3.1 Polynomials

Original: We ﬁrst deﬁne the Hermite basis for standard d -dimensional Gaussian distribution N (0 , I d ) ascomplete orthonormal for L 2 ( R d , N (0 , I d )) analysis.
Revised: We ﬁrst deﬁne the Hermite basis for standard d -dimensional Gaussian distribution N (0 , I d ) ascomplete orthonormal for L 2 ( R d , N (0 , I d )) analysis.

Original: Deﬁnition 4 (Hermite basis) .
Revised: Deﬁnition 4 (Hermite basis) .

Original: For any natural number a 2 N and x 2 R , the univariate Hermitepolynomial is deﬁned as He a ( x ) =
Revised: For any natural number a 2 N and x 2 R , the univariate Hermitepolynomial is deﬁned as He a ( x ) =

Original: (  1) a e x 2 2 d a dx a e  x 2 2 , which can be extended to a multivariate
Revised: (  1) a e x 2 2 d a dx a e  x 2 2 , which can be extended to a multivariate

Original: Hermite polynomial by taking products.
Revised: Hermite polynomial by taking products.

Original: That is, for a 2 N d , x 2 R d , d -variate the Hermitepolynomial is given by He a ( x ) =
Revised: That is, for a 2 N d , x 2 R d , d -variate the Hermitepolynomial is given by He a ( x ) =

Original: Q di =

Revised: Q di =

Original: He a i x i , which is of degree k a k 1 .
Revised: He a i x i , which is of degree k a k 1 .

Original: We then deﬁne some useful polynomails.
Revised: We then deﬁne some useful polynomails.

Original: A multivariate degree- l polynomial Hom ( x ) is calledhomogeneous if it only contains monomials of degree exactly l .
Revised: A multivariate degree- l polynomial Hom ( x ) is calledhomogeneous if it only contains monomials of degree exactly l .

Original: Similarly, a multivariate degree- lpolynomial h ( x ) is called harmonic if it can be written as a linear combination of multivariate Hermitepolynomials of degree exactly l .
Revised: Similarly, a multivariate degree- lpolynomial h ( x ) is called harmonic if it can be written as a linear combination of multivariate Hermitepolynomials of degree exactly l .

Original: Formally, we have the following deﬁnitions.
Revised: Formally, we have the following deﬁnitions.

Original: Deﬁnition 5 (Homogeneous and harmonic polynomials) .
Revised: Deﬁnition 5 (Homogeneous and harmonic polynomials) .

Original: Let A be a symmetric order- l tensor over
Revised: Let A be a symmetric order- l tensor over

Original: R d .
Revised: R d .

Original: We have
Revised: We have

Original: [Equation]
Revised: [Equation]

Original: where c j ( i 1 , . .
Revised: where c j ( i 1 , . .

Original: , i l ) denotes the total times that j appears in { i 1 , · · · , i l } .
Revised: , i l ) denotes the total times that j appears in { i 1 , · · · , i l } .

Original: Moreover, if the polynomial has at most  monomials and only applies on a sparse index set⌦ ✓ [ d ] , | ⌦ |  of variable x , i.e. p ( x ) = p ( x ⌦ ) , we say that such a polynomial is sparse.
Revised: Moreover, if the polynomial has at most  monomials and only applies on a sparse index set⌦ ✓ [ d ] , | ⌦ |  of variable x , i.e. p ( x ) = p ( x ⌦ ) , we say that such a polynomial is sparse.

Original: Deﬁnition 6 ( (  , ) -sparse polynomial) .
Revised: Deﬁnition 6 ( (  , ) -sparse polynomial) .

Original: Consider a polynomial function p : R d !
Revised: Consider a polynomial function p : R d !

Original: R that has atmost  monomials and effective coordinates.
Revised: R that has atmost  monomials and effective coordinates.

Original: We call such a polynomial a (  , ) -sparse polynomial.
Revised: We call such a polynomial a (  , ) -sparse polynomial.

Original: For a polynomial function p : R d !
Revised: For a polynomial function p : R d !

Original: R , we will denote by p ( T ) the random variable p ( x ) where x isdrawn uniformly from T .
Revised: R , we will denote by p ( T ) the random variable p ( x ) where x isdrawn uniformly from T .

Original: Moreover, the L 2 -norm of p reads k p k 2 :=
Revised: Moreover, the L 2 -norm of p reads k p k 2 :=

Original: [Equation]
Revised: [Equation]

Original: 3.2 Representative and Good Sets
Revised: 3.2 Representative and Good Sets

Original: Let ↵ 2 (0 , 12 ] be the fraction of good samples in the contaminated dataset T .
Revised: Let ↵ 2 (0 , 12 ] be the fraction of good samples in the contaminated dataset T .

Original: We say that the set ofgood samples is representative if it non-trivially represents the underlying distribution N ( µ, I d ) .
Revised: We say that the set ofgood samples is representative if it non-trivially represents the underlying distribution N ( µ, I d ) .

Original: Deﬁnition 7 (Representative set) .
Revised: Deﬁnition 7 (Representative set) .

Original: For ↵ 2 (0 , 12 ] and ⌧ 2 (0 , 1) , we say that a set S ⇢ R d isrepresentative (with respect to G ) if for any degree- l , l  2 ` , ( k 2 ` , 2 ` k 2 ` ) -sparse polynomial functionp , it holds that
Revised: For ↵ 2 (0 , 12 ] and ⌧ 2 (0 , 1) , we say that a set S ⇢ R d isrepresentative (with respect to G ) if for any degree- l , l  2 ` , ( k 2 ` , 2 ` k 2 ` ) -sparse polynomial functionp , it holds that

Original: [Equation]
Revised: [Equation]

Original: One basic requirement for a set T being ↵ -good is that at least an ↵ -fraction of the samples are from
Revised: One basic requirement for a set T being ↵ -good is that at least an ↵ -fraction of the samples are from

Original: G , i.e. 2 S .
Revised: G , i.e. 2 S .

Original: Apart from that, during the process of multiﬁltering, we do not want to remove toomany samples from S to keep S \ T representative.
Revised: Apart from that, during the process of multiﬁltering, we do not want to remove toomany samples from S to keep S \ T representative.

Original: For technical reasons, we also require that all thesamples within one subset are contained in a small L 1 ball of radius O p log( d | S | / ↵ )  .
Revised: For technical reasons, we also require that all thesamples within one subset are contained in a small L 1 ball of radius O p log( d | S | / ↵ )  .

Original: Deﬁnition 8 ( ↵ -good set) .
Revised: Deﬁnition 8 ( ↵ -good set) .

Original: For ↵ 2 (0 , 12 ] , a multiset T ⇢ R d is ↵ -good with respect to G if it holdsthat1.
Revised: For ↵ 2 (0 , 12 ] , a multiset T ⇢ R d is ↵ -good with respect to G if it holdsthat1.

Original: There exists a set S ⇢ R n which is representative (with respect to G ).
Revised: There exists a set S ⇢ R n which is representative (with respect to G ).

Original: Moreover, | S \ T | max
Revised: Moreover, | S \ T | max

Original: [Equation]
Revised: [Equation]

Original:  ↵
Revised:  ↵

Original: [Equation]
Revised: [Equation]

Original: |
Revised: |

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: We show that it is with high probability that a sufﬁcient large set S of independent samples drawnfrom N ( µ, I d ) is representative.
Revised: We show that it is with high probability that a sufﬁcient large set S of independent samples drawnfrom N ( µ, I d ) is representative.

Original: The proof is deferred in the appendix.
Revised: The proof is deferred in the appendix.

Original: Proposition 9 (Sample Complexity) .
Revised: Proposition 9 (Sample Complexity) .

Original: For ↵ 2 (0 , 12 ] and ⌧ 2 (0 , 1) , let S be a set of N =
Revised: For ↵ 2 (0 , 12 ] and ⌧ 2 (0 , 1) , let S be a set of N =

Original: O ⇣
Revised: O ⇣

Original: [Equation]
Revised: [Equation]

Original: ⌘ samples from N ( µ, I d ) .
Revised: ⌘ samples from N ( µ, I d ) .

Original: Then, S is representative(with respect to G ) with probability 1  ⌧ .
Revised: Then, S is representative(with respect to G ) with probability 1  ⌧ .

Original: 3.3 Clustering for the Initial List
Revised: 3.3 Clustering for the Initial List

Original: We will do a preliminary step of clustering that splits the input set T into an initial list of subsetsamong which at least one is ↵ -good due to Deﬁnition 8.
Revised: We will do a preliminary step of clustering that splits the input set T into an initial list of subsetsamong which at least one is ↵ -good due to Deﬁnition 8.

Original: We ﬁrst claim that the good samples, i.e.x 2 S drawn from N ( µ, I d ) , have bounded L 1 norm with high probability.
Revised: We ﬁrst claim that the good samples, i.e.x 2 S drawn from N ( µ, I d ) , have bounded L 1 norm with high probability.

Original: Claim 10 (Bounded L 1 -norm) .
Revised: Claim 10 (Bounded L 1 -norm) .

Original: For any x 2 S , we have that k x  µ k 1  p
Revised: For any x 2 S , we have that k x  µ k 1  p

Original: [Equation]
Revised: [Equation]

Original: 100 d | S | / ↵ ) 197 with probability at least ↵ / 100 .
Revised: 100 d | S | / ↵ ) 197 with probability at least ↵ / 100 .

Original: We cluster the dataset due to the bounded L 1 norm of the Gaussian samples.
Revised: We cluster the dataset due to the bounded L 1 norm of the Gaussian samples.

Original: Since | S | =
Revised: Since | S | =

Original: O  poly( k ↵ ) · poly log( dk ⌧ )  , we requires each subset T i has L 1 diameter of O  log  dk ↵ log( 1 ⌧ )  .
Revised: O  poly( k ↵ ) · poly log( dk ⌧ )  , we requires each subset T i has L 1 diameter of O  log  dk ↵ log( 1 ⌧ )  .

Original: Algorithm
Revised: Algorithm

Original: C LUSTER Input: dataset T ⇢ R d , parameter ↵ > 0 .
Revised: C LUSTER Input: dataset T ⇢ R d , parameter ↵ > 0 .

Original: [Equation]
Revised: [Equation]

Original: For each x 2 T , proceed as follows: If there are ↵ · | T | samples y in T that satisfy
Revised: For each x 2 T , proceed as follows: If there are ↵ · | T | samples y in T that satisfy

Original: [Equation]
Revised: [Equation]

Original: and no sample x 0 2 C has k x  x 0 k 1  6 C · p  , then C C [ x . 3.
Revised: and no sample x 0 2 C has k x  x 0 k 1  6 C · p  , then C C [ x . 3.

Original: For each x i 2 C , let T i
Revised: For each x i 2 C , let T i

Original: =
Revised: =

Original: { y 2 T : k x i  y k 1  6 C · p  } 4.
Revised: { y 2 T : k x i  y k 1  6 C · p  } 4.

Original: Return the list of T i ’s.
Revised: Return the list of T i ’s.

Original: We show that if T is of size at least 1 / ↵ times the sample complexity in Proposition 9, by feeding itto Algorithm 1 we obtain at least one ↵ -good subset T i in the initial list in view of Deﬁnition 8.
Revised: We show that if T is of size at least 1 / ↵ times the sample complexity in Proposition 9, by feeding itto Algorithm 1 we obtain at least one ↵ -good subset T i in the initial list in view of Deﬁnition 8.

Original: Lemma (Initial
Revised: Lemma (Initial

Original: List) .
Revised: List) .

Original: Given ↵ (0 , 12 ] and a set
Revised: Given ↵ (0 , 12 ] and a set

Original: T of
Revised: T of

Original: O
Revised: O

Original: [Equation]
Revised: [Equation]

Original: ⌘ samples in R d of which a 2 ↵ -fraction areindependent samples from N ( µ, I d ) , Algorithm 1 returns a list of at most 1 / ↵ many subsets of T ,such that with probability at least 1  ⌧ , there exists at least one of them is ↵ -good with respect to G .
Revised: ⌘ samples in R d of which a 2 ↵ -fraction areindependent samples from N ( µ, I d ) , Algorithm 1 returns a list of at most 1 / ↵ many subsets of T ,such that with probability at least 1  ⌧ , there exists at least one of them is ↵ -good with respect to G .

Original: The motivation of bounding the L 1 -distance is to make sure the function value of any ( k 2 ` , 2 ` k 2 ` ) -sparse p is bounded for samples in the ↵ -good subset T i (technically, also assuming Var[ p ( G )]  1 ).
Revised: The motivation of bounding the L 1 -distance is to make sure the function value of any ( k 2 ` , 2 ` k 2 ` ) -sparse p is bounded for samples in the ↵ -good subset T i (technically, also assuming Var[ p ( G )]  1 ).

Original: This is because when there exist a signiﬁcant fraction of good samples in T i , we want to efﬁcientlydistinguish the inliers from the outliers.
Revised: This is because when there exist a signiﬁcant fraction of good samples in T i , we want to efﬁcientlydistinguish the inliers from the outliers.

Original: A value-bounded polynomial function will facilitate ouranalysis of the function variance.
Revised: A value-bounded polynomial function will facilitate ouranalysis of the function variance.

Original: Speciﬁcally, we have the following guarantee.
Revised: Speciﬁcally, we have the following guarantee.

Original: Lemma 12.
Revised: Lemma 12.

Original: Suppose T is ↵ -good and for some ( k 2 ` , 2 ` k 2 ` ) -sparse p of degree l , we have
Revised: Suppose T is ↵ -good and for some ( k 2 ` , 2 ` k 2 ` ) -sparse p of degree l , we have

Original: Var[ p ( G )]  1 .
Revised: Var[ p ( G )]  1 .

Original: Then, it holds that max x,y 2 T | p ( x )  p ( y ) |  O   2 ` · k 2 ` · log  dk ↵ log  1 ⌧  l/ .
Revised: Then, it holds that max x,y 2 T | p ( x )  p ( y ) |  O   2 ` · k 2 ` · log  dk ↵ log  1 ⌧  l/ .

Original:  The Multiﬁltering Algorithm
Revised:  The Multiﬁltering Algorithm

Original: We start with introducing the multiﬁltering framework that is broadly used in prior works [DKS18b,
Revised: We start with introducing the multiﬁltering framework that is broadly used in prior works [DKS18b,

Original: DKK20a, DKK + 21b], followed by our new techniques and algorithmic designs for learning thesparse true mean with attribute-efﬁcient performance.
Revised: DKK20a, DKK + 21b], followed by our new techniques and algorithmic designs for learning thesparse true mean with attribute-efﬁcient performance.

Original: The multiﬁltering framework, i.e. Algorithm 2, includes three main steps.
Revised: The multiﬁltering framework, i.e. Algorithm 2, includes three main steps.

Original: The ﬁrst one is our initialclustering step that generates the intial list L with guarantees in Lemma 11.
Revised: The ﬁrst one is our initialclustering step that generates the intial list L with guarantees in Lemma 11.

Original: We then consider a treeof subsets, of which the root is the original contaminated sample set T and each child node of theroot represents a subset in L .
Revised: We then consider a treeof subsets, of which the root is the original contaminated sample set T and each child node of theroot represents a subset in L .

Original: The multiﬁlter then iterates through the subsets and does one of thefollowing: (1) ﬁlter the set by removing outliers and create one child node for it, (2) further splitthe set into two potentially overlapping subsets and create two children nodes, (3) certify that theset can’t be good and delete branch.
Revised: The multiﬁlter then iterates through the subsets and does one of thefollowing: (1) ﬁlter the set by removing outliers and create one child node for it, (2) further splitthe set into two potentially overlapping subsets and create two children nodes, (3) certify that theset can’t be good and delete branch.

Original: In the end, if all leaves of the tree can not be further ﬁltered,split or deleted, the mean of the subsets on leaf nodes will be returned as a list M .
Revised: In the end, if all leaves of the tree can not be further ﬁltered,split or deleted, the mean of the subsets on leaf nodes will be returned as a list M .

Original: The ﬁnal step is ablack-box list reduction algorithm that reduces the list size of M from poly(1 / ↵ ) to O (1 / ↵ ) .
Revised: The ﬁnal step is ablack-box list reduction algorithm that reduces the list size of M from poly(1 / ↵ ) to O (1 / ↵ ) .

Original: the list { T i , ↵
Revised: the list { T i , ↵

Original: T
Revised: T

Original: LUSTER
Revised: LUSTER

Original: empty list.
Revised: empty list.

Original: is non
Revised: is non

Original: empty
Revised: empty

Original: (a) Get the ﬁrst element ( T 0 , ↵ 0 ) from L and remove it from the list.
Revised: (a) Get the ﬁrst element ( T 0 , ↵ 0 ) from L and remove it from the list.

Original: EFFICIENT
Revised: EFFICIENT

Original: do one of the following: (i)
Revised: do one of the following: (i)

Original: If the algorithm returns a vector, add it to M .
Revised: If the algorithm returns a vector, add it to M .

Original: (ii)
Revised: (ii)

Original: If it returns a list of ( T i , ↵ i ) , remove those with ↵ i > 1 , and append the remainings to L .
Revised: If it returns a list of ( T i , ↵ i ) , remove those with ↵ i > 1 , and append the remainings to L .

Original: (iii)
Revised: (iii)

Original: If it returns “NO”, proceeds to the next iteration.
Revised: If it returns “NO”, proceeds to the next iteration.

Original: Run L IST R EDUCTION to the list M and return its output.
Revised: Run L IST R EDUCTION to the list M and return its output.

Original: In the following sections, we will elaborate on our design of the A TTRIBUTE - EFFICIENTM ULTIFILTER , which admits the desired guarantees as in Theorem 1.
Revised: In the following sections, we will elaborate on our design of the A TTRIBUTE - EFFICIENTM ULTIFILTER , which admits the desired guarantees as in Theorem 1.

Original: The black-box list reductionalgorithm follows directly from [DKS18b].
Revised: The black-box list reductionalgorithm follows directly from [DKS18b].

Original: We will include in the appendix for reader’s reference.
Revised: We will include in the appendix for reader’s reference.

Original: 4.1 The Attribute-efﬁcient Multiﬁlter
Revised: 4.1 The Attribute-efﬁcient Multiﬁlter

Original: To achieve attribute-efﬁcient multiﬁltering, the main multiﬁlter only works on a few attributes ofthe samples at one time.
Revised: To achieve attribute-efﬁcient multiﬁltering, the main multiﬁlter only works on a few attributes ofthe samples at one time.

Original: That is, it ﬁxes an ⌦ ✓ [ d ] and then only looks at x ⌦ , 8 x 2 T .
Revised: That is, it ﬁxes an ⌦ ✓ [ d ] and then only looks at x ⌦ , 8 x 2 T .

Original: For thispurpose, Algorithm 3 searches for a sufﬁcient condition of the samples being concentrated such thatthe empirical mean is a close estimation.
Revised: For thispurpose, Algorithm 3 searches for a sufﬁcient condition of the samples being concentrated such thatthe empirical mean is a close estimation.

Original: Precisely, observing that the eigenvalue of an empiricalcovariance matrix is exactly the variance of the samples along with its corresponding eigenvector, bysearching the largest eigenvalue over all k -sparse support sets we can certify whether there still existoutliers in the sample set.
Revised: Precisely, observing that the eigenvalue of an empiricalcovariance matrix is exactly the variance of the samples along with its corresponding eigenvector, bysearching the largest eigenvalue over all k -sparse support sets we can certify whether there still existoutliers in the sample set.

Original: Input: multiset T ⇢ R d , parameter ↵ .
Revised: Input: multiset T ⇢ R d , parameter ↵ .

Original: Let P d, ` ( x ) be the vector of all degree- ` Hermite basis, i.e.
Revised: Let P d, ` ( x ) be the vector of all degree- ` Hermite basis, i.e.

Original: [Equation]
Revised: [Equation]

Original: Calculate sample mean ˜ µ = E ( T ) , and the degree- ` “covariance matrix” ˜
Revised: Calculate sample mean ˜ µ = E ( T ) , and the degree- ` “covariance matrix” ˜

Original: [Equation]
Revised: [Equation]

Original: P
Revised: P

Original: [Equation]
Revised: [Equation]

Original: Let C > 0 be a sufﬁciently large universal constant.
Revised: Let C > 0 be a sufﬁciently large universal constant.

Original: . Find
Revised: . Find

Original: k
Revised: k

Original: 2 ` entries in ˜ ⌃ with the
Revised: 2 ` entries in ˜ ⌃ with the

Original: largest
Revised: largest

Original: magnitude
Revised: magnitude

Original: of which
Revised: of which

Original: k `
Revised: k `

Original: are on the diagonal and k
Revised: are on the diagonal and k

Original: ` 
Revised: `

Original: k ` are
Revised: k ` are

Original: off
Revised: off

Original: -diagonal.
Revised: -diagonal.

Original: Let U ✓ [ d ` ] ⇥
Revised: Let U ✓ [ d ` ] ⇥

Original: [ d ` ] be
Revised: [ d ` ] be

Original: the index set of
Revised: the index set of

Original: these k 2 ` entries.
Revised: these k 2 ` entries.

Original: Since ˜ ⌃ is symmetric,
Revised: Since ˜ ⌃ is symmetric,

Original: U will be selected such that the (˜ ⌃ ) U is symmetric as well.
Revised: U will be selected such that the (˜ ⌃ ) U is symmetric as well.

Original: If
Revised: If

Original: [Equation]
Revised: [Equation]

Original: ) 2 ` , then return ˆ µ = hard k (˜ µ ) .
Revised: ) 2 ` , then return ˆ µ = hard k (˜ µ ) .

Original: Set U 

Revised: Set U

Original: = { ( i, j ) 

Revised: = { ( i, j )

Original: [ d ` ] ⇥ [ d ` ] : 9 ( i, j 0 ) 2 U and ( i 0 , j ) 2 U } .
Revised: [ d ` ] ⇥ [ d ` ] : 9 ( i, j 0 ) 2 U and ( i 0 , j ) 2 U } .

Original: Compute largest eigenvalue  ⇤ of (˜ ⌃ ) U 0 , and its corresponding eigenvector v ⇤ .
Revised: Compute largest eigenvalue  ⇤ of (˜ ⌃ ) U 0 , and its corresponding eigenvector v ⇤ .

Original: If  ⇤  C · (( ` + C log(
Revised: If  ⇤  C · (( ` + C log(

Original: 1 ↵
Revised: 1 ↵

Original: )) log 2 (
Revised: )) log 2 (

Original: 2 + log( 1 ↵
Revised: 2 + log( 1 ↵

Original: ) 2 ` , then apply
Revised: ) 2 ` , then apply

Original: i) B ASIC
Revised: i) B ASIC

Original: M ULTIFILTER if ` = 1 , or (
Revised: M ULTIFILTER if ` = 1 , or (

Original: ii
Revised: ii

Original: ) H ARMONIC M ULTIFILTER if ` >
Revised: ) H ARMONIC M ULTIFILTER if ` >

Original: , on the
Revised: , on the

Original: degree- ` polynomial p 1 ( x ) = v ⇤ · P d, ` ( x )
Revised: degree- ` polynomial p 1 ( x ) = v ⇤ · P d, ` ( x )

Original: Otherwise , let
Revised: Otherwise , let

Original: A 0 = (˜ ⌃ ) U
Revised: A 0 = (˜ ⌃ ) U

Original: , and
Revised: , and

Original: the normalized A =
Revised: the normalized A =

Original: Ak A 0 k F
Revised: Ak A 0 k F

Original: Let
Revised: Let

Original: p 2 ( x ) =
Revised: p 2 ( x ) =

Original: k A 0 k F · 
Revised: k A 0 k F ·

Original: P d, ` ( x ) >
Revised: P d, ` ( x ) >

Original: A 0 P d, ` ( x )  =
Revised: A 0 P d, ` ( x )  =

Original: k
Revised: k

Original: A 0 k F · p A 0 ( x ) be the degree- 2 ` harmonic function, run H ARMONIC M ULTIFILTER on p 2 ( x ) .
Revised: A 0 k F · p A 0 ( x ) be the degree- 2 ` harmonic function, run H ARMONIC M ULTIFILTER on p 2 ( x ) .

Original: Unfortunately, searching for the largest eigenvalue over k -sparse support sets is a NP-hard problem,which requires iterating over all possible combinations.
Revised: Unfortunately, searching for the largest eigenvalue over k -sparse support sets is a NP-hard problem,which requires iterating over all possible combinations.

Original: We instead use the Frobenius norm which upper bounds the largest eigenvalue, i.e. the spectral norm, of the submatrices.
Revised: We instead use the Frobenius norm which upper bounds the largest eigenvalue, i.e. the spectral norm, of the submatrices.

Original: That is, we willsimply ﬁnd an index set U which contains the largest k 2 ` entries of ˜ ⌃ (in magnitude).
Revised: That is, we willsimply ﬁnd an index set U which contains the largest k 2 ` entries of ˜ ⌃ (in magnitude).

Original: If all these k 2 `entries are small, then the Frobenius and spectral norms of any k ` ⇥ k ` submatrices of ˜ ⌃ are alsobounded, meaning that the samples behave like a standard Gaussian.
Revised: If all these k 2 `entries are small, then the Frobenius and spectral norms of any k ` ⇥ k ` submatrices of ˜ ⌃ are alsobounded, meaning that the samples behave like a standard Gaussian.

Original: As a result, if at Step 3, thealgorithm returns a vector ˆ µ , we can certify that it is sufﬁciently close to the true mean µ .
Revised: As a result, if at Step 3, thealgorithm returns a vector ˆ µ , we can certify that it is sufﬁciently close to the true mean µ .

Original: Lemma 13.
Revised: Lemma 13.

Original: If T is ↵ -good, when Algorithm 3 returns a vector ˆ µ = hard k (˜ µ ) at Step 3.
Revised: If T is ↵ -good, when Algorithm 3 returns a vector ˆ µ = hard k (˜ µ ) at Step 3.

Original: , we havethat k µ
Revised: , we havethat k µ

Original: [Equation]
Revised: [Equation]

Original: O
Revised: O

Original: [Equation]
Revised: [Equation]

Original: ⌘ .
Revised: ⌘ .

Original: On the other hand, if Algorithm 3 does not return at Step 3, we show that either Step 6 or Step 7 wouldmake progress such that the algorithm can proceed.
Revised: On the other hand, if Algorithm 3 does not return at Step 3, we show that either Step 6 or Step 7 wouldmake progress such that the algorithm can proceed.

Original: First, if the largest eigenvalue  ⇤ calculated in
Revised: First, if the largest eigenvalue  ⇤ calculated in

Original: Step 5 is sufﬁciently large, the algorithm ﬁnds a direction in which the outliers are distinguishablefrom the inliers.
Revised: Step 5 is sufﬁciently large, the algorithm ﬁnds a direction in which the outliers are distinguishablefrom the inliers.

Original: This sufﬁces for the basic multiﬁlter to proceed, i.e. it does not return “YES”.
Revised: This sufﬁces for the basic multiﬁlter to proceed, i.e. it does not return “YES”.

Original: Inother words, the basic multiﬁlter must either certify that T is not ↵ -good, or return a list of subsets ofwhich at least one is cleaner than the input set.
Revised: Inother words, the basic multiﬁlter must either certify that T is not ↵ -good, or return a list of subsets ofwhich at least one is cleaner than the input set.

Original: In either way, the algorithm can proceed.
Revised: In either way, the algorithm can proceed.

Original: Lemma 14 (Degree- ` Harmonic Polynomial) .
Revised: Lemma 14 (Degree- ` Harmonic Polynomial) .

Original: If Algorithm 3 ﬁnds that Var[ v ⇤ · T ] =  ⇤ 
Revised: If Algorithm 3 ﬁnds that Var[ v ⇤ · T ] =  ⇤

Original: C · (( ` + C log( 1 ↵ )) log 2 (2 + log( 1 ↵ )))
Revised: C · (( ` + C log( 1 ↵ )) log 2 (2 + log( 1 ↵ )))

Original: 2 ` , Algorithm 4 must not return “YES” at Step 4(a).
Revised: 2 ` , Algorithm 4 must not return “YES” at Step 4(a).

Original: The step of ﬁltering by the degree- ` harmonic polynomial essentially relies on the condition thatthe algorithm can ﬁnd a sufﬁciently large eigenvalue  ⇤ .
Revised: The step of ﬁltering by the degree- ` harmonic polynomial essentially relies on the condition thatthe algorithm can ﬁnd a sufﬁciently large eigenvalue  ⇤ .

Original: However, the condition can be brokenif  ⇤ is much less than  (˜ ⌃  I ) U 
Revised: However, the condition can be brokenif  ⇤ is much less than  (˜ ⌃  I ) U

Original: F , since we use a sufﬁcient but not necessary condition at
Revised: F , since we use a sufﬁcient but not necessary condition at

Original: Step 3.
Revised: Step 3.

Original: Fortunately, if the algorithm does not ﬁnd a large  ⇤ , we show that it can still proceed bycalling the harmonic multiﬁlter on a degree- 2 ` polynomial.
Revised: Fortunately, if the algorithm does not ﬁnd a large  ⇤ , we show that it can still proceed bycalling the harmonic multiﬁlter on a degree- 2 ` polynomial.

Original: This is because the degree- 2 ` (originallyquadratic when ` = 1 ) polynomials are more sensitive to a small amount of outliers than the degree- `polynomials [DKS18b, DKK + 19].
Revised: This is because the degree- 2 ` (originallyquadratic when ` = 1 ) polynomials are more sensitive to a small amount of outliers than the degree- `polynomials [DKS18b, DKK + 19].

Original: As a result, we will have the following guarantee.
Revised: As a result, we will have the following guarantee.

Original: Lemma 15 (Degree- 2 ` Harmonic Polynomial) .
Revised: Lemma 15 (Degree- 2 ` Harmonic Polynomial) .

Original: If Algorithm 3 neither returns at Step 3, nor ﬁlters at
Revised: If Algorithm 3 neither returns at Step 3, nor ﬁlters at

Original: Step 6, then it must proceed with the harmonic multiﬁlter in Step 7.
Revised: Step 6, then it must proceed with the harmonic multiﬁlter in Step 7.

Original: Notice that p 1 and p 2 can both be of degree greater than 1 .
Revised: Notice that p 1 and p 2 can both be of degree greater than 1 .

Original: The algorithmic design for higher-degreepolynomials is more subtle due to two factors.
Revised: The algorithmic design for higher-degreepolynomials is more subtle due to two factors.

Original: The ﬁrst is that the basic multiﬁlter requires the inputpolynomial p have a small variance on G , i.e. Var[ p ( G )]  1 , when T is ↵ -good.
Revised: The ﬁrst is that the basic multiﬁlter requires the inputpolynomial p have a small variance on G , i.e. Var[ p ( G )]  1 , when T is ↵ -good.

Original: For this reason,
Revised: For this reason,

Original: Algorithm 5 utilizes a set of multilinear polynomials to verify the variance of quadratic harmonicpolynomials.
Revised: Algorithm 5 utilizes a set of multilinear polynomials to verify the variance of quadratic harmonicpolynomials.

Original: This leads out our second idea of design that the polynomial function p used to ﬁltermust consist of only Hermite basis.
Revised: This leads out our second idea of design that the polynomial function p used to ﬁltermust consist of only Hermite basis.

Original: More details can be found in Section 4.3.
Revised: More details can be found in Section 4.3.

Original: 4.2 Basic Multiﬁlter for Sparse Polynomials
Revised: 4.2 Basic Multiﬁlter for Sparse Polynomials

Original: The basic multiﬁlter, which is a key ingredient in the multiﬁltering framework, takes as input a( k 2 ` , 2 ` k 2 ` ) -sparse polynomial p and use it to certify whether T is ↵ -good and sufﬁciently concentrated.
Revised: The basic multiﬁlter, which is a key ingredient in the multiﬁltering framework, takes as input a( k 2 ` , 2 ` k 2 ` ) -sparse polynomial p and use it to certify whether T is ↵ -good and sufﬁciently concentrated.

Original: The central idea is to measure how p ( T ) distributed and compare it to that of the Gaussianvariable G .
Revised: The central idea is to measure how p ( T ) distributed and compare it to that of the Gaussianvariable G .

Original: Therefore, we require the input p has small variance on G , i.e. Var[ p ( G )]
Revised: Therefore, we require the input p has small variance on G , i.e. Var[ p ( G )]

Original:  1 , asotherwise, it could ﬁlter away a large quantity of the good samples.
Revised:  1 , asotherwise, it could ﬁlter away a large quantity of the good samples.

Original: Notice that when p is linear, thisrequirement is satisﬁed due to the property of Guassian.
Revised: Notice that when p is linear, thisrequirement is satisﬁed due to the property of Guassian.

Original: When p is of higher degree, we will invokethe harmonic multiﬁlter which includes additional veriﬁcation steps.
Revised: When p is of higher degree, we will invokethe harmonic multiﬁlter which includes additional veriﬁcation steps.

Original: Given that p is sufﬁciently concentrated on Gaussian variable G , it remains to ﬁnd the appropriatethreshold at Step 4(b)(i) for ﬁltering.
Revised: Given that p is sufﬁciently concentrated on Gaussian variable G , it remains to ﬁnd the appropriatethreshold at Step 4(b)(i) for ﬁltering.

Original: An appropriate threshold is which the algorithm can use it toﬁlter a large amount of the outliers in T , but keeps most of the samples from the representative set S .
Revised: An appropriate threshold is which the algorithm can use it toﬁlter a large amount of the outliers in T , but keeps most of the samples from the representative set S .

Original: For this purpose, the parameter in Step 4(b)(i) is set to be 8 / ↵ times larger than the concentrationbound of S (due to Deﬁnition 7 and the concentration of Gaussian variables).
Revised: For this purpose, the parameter in Step 4(b)(i) is set to be 8 / ↵ times larger than the concentrationbound of S (due to Deﬁnition 7 and the concentration of Gaussian variables).

Original: In other words, eachtime Step 4(b) is executed, the number of the samples removed from T is 8 / ↵ times more than thatof the good samples in S , which means most of the removed samples are outliers.
Revised: In other words, eachtime Step 4(b) is executed, the number of the samples removed from T is 8 / ↵ times more than thatof the good samples in S , which means most of the removed samples are outliers.

Original: As a result, the very novel analysis here is showing that if Algorithm 4 reaches Step 4(b)(i), theremust exist a threshold t such that the algorithm can proceed.
Revised: As a result, the very novel analysis here is showing that if Algorithm 4 reaches Step 4(b)(i), theremust exist a threshold t such that the algorithm can proceed.

Original: The idea is to show by contradiction thatif such threshold does not exist, the set T must be sufﬁciently concentrated such that the algorithmwill return at Step 4(a).
Revised: The idea is to show by contradiction thatif such threshold does not exist, the set T must be sufﬁciently concentrated such that the algorithmwill return at Step 4(a).

Original: This essentially relies on our result of the initial clustering of Algorithm 1,which guarantees that each subset T is bounded in a small ` 1 ball and the function value of p on the↵ -good T does not change drastically (Lemma 12).
Revised: This essentially relies on our result of the initial clustering of Algorithm 1,which guarantees that each subset T is bounded in a small ` 1 ball and the function value of p on the↵ -good T does not change drastically (Lemma 12).

Original: As a result, we have the following guarantees for
Revised: As a result, we have the following guarantees for

Original: Algorithm 4.
Revised: Algorithm 4.

Original: The detailed proof is deferred to the appendix.
Revised: The detailed proof is deferred to the appendix.

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: / 100)and | T |  max( | T 1 | , | T 2 | )  ↵ | T | / 4 .
Revised: / 100)and | T |  max( | T 1 | , | T 2 | )  ↵ | T | / 4 .

Original: (ii) Let ↵ i =
Revised: (ii) Let ↵ i =

Original: ↵ · (1  ↵ 2 / 100) · | T | / | T i | , for i = 1 , 2 .
Revised: ↵ · (1  ↵ 2 / 100) · | T | / | T i | , for i = 1 , 2 .

Original: (iii) Return { ( T 1 , ↵ 1 ) , ( T 2 , ↵ 2 ) } .
Revised: (iii) Return { ( T 1 , ↵ 1 ) , ( T 2 , ↵ 2 ) } .

Original: Proposition 16 (Basic Multiﬁlter) .
Revised: Proposition 16 (Basic Multiﬁlter) .

Original: Given a set T ⇢ R d , parameter 0 < ↵  1 , and a 292 degree- l , ( k 2 ` , 2 ` k 2 ` ) -sparse polynomial p : R d !
Revised: Given a set T ⇢ R d , parameter 0 < ↵  1 , and a 292 degree- l , ( k 2 ` , 2 ` k 2 ` ) -sparse polynomial p : R d !

Original: R for which it holds that T is ↵ -good then
Revised: R for which it holds that T is ↵ -good then

Original: Var[ p ( G )]  1 , Algorithm 4 returns one of the following with guarantees: (1) “YES”, and wehave that (i)
Revised: Var[ p ( G )]  1 , Algorithm 4 returns one of the following with guarantees: (1) “YES”, and wehave that (i)

Original: Var
Revised: Var

Original: [ p ( T )]  O  (2 ` + log( 1 ↵ ))
Revised: [ p ( T )]  O  (2 ` + log( 1 ↵ ))

Original: l · log 2 (2 + log( 1 ↵ ))  , and (ii)
Revised: l · log 2 (2 + log( 1 ↵ ))  , and (ii)

Original: If T is ↵ -good, then| E [ p ( G )]  E [ p ( T )] |  O  (
Revised: If T is ↵ -good, then| E [ p ( G )]  E [ p ( T )] |  O  (

Original: [Equation]
Revised: [Equation]

Original: + log( 1 ↵ ))
Revised: + log( 1 ↵ ))

Original: l 2 · log(2 + log( 1 ↵ ))  .
Revised: l 2 · log(2 + log( 1 ↵ ))  .

Original: (2) “NO”, then T is not ↵ -good.
Revised: (2) “NO”, then T is not ↵ -good.

Original: A list of pairs { ( T i , ↵ i ) } , T i ⇢ T , of length one or two satisfying the multiﬁlter condition for ( T, ↵ ) .
Revised: A list of pairs { ( T i , ↵ i ) } , T i ⇢ T , of length one or two satisfying the multiﬁlter condition for ( T, ↵ ) .

Original: 4.3 Harmonic Multiﬁlter with Hermite Basis
Revised: 4.3 Harmonic Multiﬁlter with Hermite Basis

Original: The purpose of having Step 7 in the main multiﬁlter is to proceed even when no degree- ` polynomialis found to have large variance on the set of samples.
Revised: The purpose of having Step 7 in the main multiﬁlter is to proceed even when no degree- ` polynomialis found to have large variance on the set of samples.

Original: This step is necessary because even though thelargest eigenvalue  ⇤ on the current index set U is small, it does not necessarily mean that it is smallon all k 2 ` -sparse index sets.
Revised: This step is necessary because even though thelargest eigenvalue  ⇤ on the current index set U is small, it does not necessarily mean that it is smallon all k 2 ` -sparse index sets.

Original: Thereby, it is of most importance that Step 7 of Algorithm 3 inevitablyﬁlters the set T or certify that T is not ↵ -good given the Frobenius norm of (˜ ⌃ )
Revised: Thereby, it is of most importance that Step 7 of Algorithm 3 inevitablyﬁlters the set T or certify that T is not ↵ -good given the Frobenius norm of (˜ ⌃ )

Original: U is large.
Revised: U is large.

Original: For thisreason, we construct another sparse harmonic polynomial p 2 ( x ) which can be seens as “projecting”
Revised: For thisreason, we construct another sparse harmonic polynomial p 2 ( x ) which can be seens as “projecting”

Original: P d, ` ( x  ˜ µ ) P d, ` ( x  ˜ µ ) > on (˜ ⌃ )
Revised: P d, ` ( x  ˜ µ ) P d, ` ( x  ˜ µ ) > on (˜ ⌃ )

Original: U for each x 2 T .
Revised: U for each x 2 T .

Original: Intuitively, the algorithm must ﬁnd some x thatcause the large deviation of ˜ ⌃ from that of Gaussian.
Revised: Intuitively, the algorithm must ﬁnd some x thatcause the large deviation of ˜ ⌃ from that of Gaussian.

Original: Thus, cleaner sample sets must be returned,except that T is veriﬁed to be not ↵ -good.
Revised: Thus, cleaner sample sets must be returned,except that T is veriﬁed to be not ↵ -good.

Original: Recall that applying the basic multiﬁlter on a polynomial p requires Var[ p ( G )]  1 .
Revised: Recall that applying the basic multiﬁlter on a polynomial p requires Var[ p ( G )]  1 .

Original: This does notcome for free for a higher-degree polynomial p (say p 2 , which is at least quadratic), as the varianceof higher-degree polynomials on the Gaussian variable essentially depends on the distribution mean,i.e. µ  ˜ µ in this case, which is unfortunately unknown.
Revised: This does notcome for free for a higher-degree polynomial p (say p 2 , which is at least quadratic), as the varianceof higher-degree polynomials on the Gaussian variable essentially depends on the distribution mean,i.e. µ  ˜ µ in this case, which is unfortunately unknown.

Original: As a remedy, notice that our p 1 and p 2 aredesigned to be harmonic in light of an observation in [DKS18b].
Revised: As a remedy, notice that our p 1 and p 2 aredesigned to be harmonic in light of an observation in [DKS18b].

Original: Roughly speaking, for any harmonicpolynomial h A ( x ) , E x ⇠ N ( µ 0 , I )
Revised: Roughly speaking, for any harmonicpolynomial h A ( x ) , E x ⇠ N ( µ 0 , I )

Original: [ h A ( x ) 2 ] is equal to the summation of homogeneous polynomials ofµ 0 , which can also be seen as the expectation of multilinear polynomials over independent variables
Revised: [ h A ( x ) 2 ] is equal to the summation of homogeneous polynomials ofµ 0 , which can also be seen as the expectation of multilinear polynomials over independent variables

Original: X ( i ) ⇠ N ( µ 0 , I ) .
Revised: X ( i ) ⇠ N ( µ 0 , I ) .

Original: Thus, we only need to verify the expectation of these corresponding multilinearpolynomials, whose variance on G does not grow with µ 0 .
Revised: Thus, we only need to verify the expectation of these corresponding multilinearpolynomials, whose variance on G does not grow with µ 0 .

Original: The harmonic multiﬁlter is presented in
Revised: The harmonic multiﬁlter is presented in

Original: Algorithm 5.
Revised: Algorithm 5.

Original: Lemma 17 (Variance of p ) .
Revised: Lemma 17 (Variance of p ) .

Original: Given multilinear polynomail A ( x 1 , . . .
Revised: Given multilinear polynomail A ( x 1 , . . .

Original: , x l ) , x 1 , . . .
Revised: , x l ) , x 1 , . . .

Original: x l 2 R dwith k
Revised: x l 2 R dwith k

Original: A k 2 = 1 , multiset T ⇢ R d , parameter ↵ , and conﬁdence ⌧ , if M ULTILINEARM ULTIFILTER returns “YES” correctly and if T is ↵ -good, then we have E [ p ( G  ˜ µ ) 2 ] =
Revised: A k 2 = 1 , multiset T ⇢ R d , parameter ↵ , and conﬁdence ⌧ , if M ULTILINEARM ULTIFILTER returns “YES” correctly and if T is ↵ -good, then we have E [ p ( G  ˜ µ ) 2 ] =

Original: O  (2 ` + log(1 / ↵ )) · log 2 (2 + log(1 / ↵ ))
Revised: O  (2 ` + log(1 / ↵ )) · log 2 (2 + log(1 / ↵ ))

Original: 2 l .
Revised:  l .

Original: Due to Lemma 17, we have that Var[ p ( G  ˜ µ ) /  ]  1 , for which we can invoke the basic multiﬁlter.
Revised: Due to Lemma 17, we have that Var[ p ( G  ˜ µ ) /  ]  1 , for which we can invoke the basic multiﬁlter.

Original: This gives our design of Algorithm 5.
Revised: This gives our design of Algorithm 5.

Original: Input: degree- l harmonic polynomial p ( x ) =
Revised: Input: degree- l harmonic polynomial p ( x ) =

Original: h A ( x ) with ( k 2 ` , 2 ` k 2 ` ) -sparse A , multiset T , parameter ↵ , conﬁdence ⌧ .
Revised: h A ( x ) with ( k 2 ` , 2 ` k 2 ` ) -sparse A , multiset T , parameter ↵ , conﬁdence ⌧ .

Original: For  0 = 0 to l : (a) Let B ( l 0 ) by the order- 2 l 0 tensor withB
Revised: For l 0 = 0 to l : (a) Let B ( l 0 ) by the order- 2 l 0 tensor with

Original: [Equation]
Revised: [Equation]

Original: (b) Consider B ( l 0 ) as an d k 0 ⌦ d k 0 symmetric matrix by grouping each of the i 1 , . .
Revised: (b) Consider B ( l 0 ) as an d k 0 ⌦ d k 0 symmetric matrix by grouping each of the i 1 , . .

Original: , i l 0 and j 1 , . .
Revised: , i l 0 and j 1 , . .

Original: , j l 0 coordinates together.
Revised: , j l 0 coordinates together.

Original: Do eigendecomposition to B ( l 0 ) such that
Revised: Do eigendecomposition to B ( l 0 ) such that

Original: [Equation]
Revised: [Equation]

Original: (c) Run M ULTILINEAR M ULTIFILTER on each ( T, V i , l 0 , ↵ , ⌧ / ( ld l )) .
Revised: (c) Run M ULTILINEAR M ULTIFILTER on each ( T, V i , l 0 , ↵ , ⌧ / ( ld l )) .

Original: If it returns “NO” or a list of ( 
Revised: If it returns “NO” or a list of ( T

Original: T, w i , ↵ ) , then return the same output.
Revised: , w i , ↵ ) , then return the same output.

Original: Run B ASIC M ULTIFILTER on p ( x  ˜ µ ) /  , where  :=
Revised: Run B ASIC M ULTIFILTER on p ( x  ˜ µ ) /  , where  :=

Revised: If it returns “NO”

Revised: or

Original: If it returns “NO” or a list of ( T i , ↵ i ) , return the same output.
Revised: a list of ( T i , ↵ i ) , return the same output If it returns

Original: If it returns “YES”, also return “NO”
Revised: “YES”, also return “NO”

Original: Lemma 18 (Harmonic Multiﬁlter Must Filter) .
Revised: Lemma 18 (Harmonic Multiﬁlter Must Filter) .

Original: If Algorithm 5 reaches Step 2 for p 2 , it either certiﬁesthat T is not ↵ -good, or returns a list of ( T i , ↵ i ) such that all T i constain less samples than T .
Revised: If Algorithm 5 reaches Step 2 for p 2 , it either certiﬁesthat T is not ↵ -good, or returns a list of ( T i , ↵ i ) such that all T i constain less samples than T .

Original: To prove Lemma 18, we only need to show that if T is ↵ -good, the basic multiﬁlter at Step 4 mustnot return “YES”.
Revised: To prove Lemma 18, we only need to show that if T is ↵ -good, the basic multiﬁlter at Step 4 mustnot return “YES”.

Original: This can be implied from the guarantee 1(ii) of Proposition 16.
Revised: This can be implied from the guarantee 1(ii) of Proposition 16.

Original: Speciﬁcally, if T is↵ -good, returning “YES” means that E [ p ( G )] is close to E [ p ( T )] .
Revised: Speciﬁcally, if T is↵ -good, returning “YES” means that E [ p ( G )] is close to E [ p ( T )] .

Original: In light of Lemma 17, | E [ p ( G )] | issmall implies that | E [ p ( T )] | is also small.
Revised: In light of Lemma 17, | E [ p ( G )] | issmall implies that | E [ p ( T )] | is also small.

Original: However, by construction we have that  (˜ ⌃ ) U 
Revised: However, by construction we have that  (˜ ⌃ ) U

Original: =
Revised: =

Original: E [ p ( T )]is large, which is a contradiction.
Revised: E [ p ( T )]is large, which is a contradiction.

Original: Thus, if the basic multiﬁlter return “YES”, we can conclude that Tis not ↵ -good and return “NO”.
Revised: Thus, if the basic multiﬁlter return “YES”, we can conclude that Tis not ↵ -good and return “NO”.

Original: This completes the proof for Lemma 15.
Revised: This completes the proof for Lemma 15.

Original:  Conclusion and Future Work
Revised:  Conclusion and Future Work

Original: In this paper, we introduce an attribute-efﬁcient mean estimation algorithm which achieves samplecomplexity polylogarithmic in dimension with low-degree sparse polynomials under the list-decodablelearning settings.
Revised: In this paper, we introduce an attribute-efﬁcient mean estimation algorithm which achieves samplecomplexity polylogarithmic in dimension with low-degree sparse polynomials under the list-decodablelearning settings.

Original: There is evidence showing that smaller error rates are achievable under the Gaussianassumption by leveraging higher-degree polynomials, which can be an interesting future direction.
Revised: There is evidence showing that smaller error rates are achievable under the Gaussianassumption by leveraging higher-degree polynomials, which can be an interesting future direction.

Original: Another natural question is whether the current techniques could be utilized to attribute-efﬁcientlysolve the other interesting learning problems such as learning of halfspaces, polynomial thresholdfunctions, and regression.
Revised: Another natural question is whether the current techniques could be utilized to attribute-efﬁcientlysolve the other interesting learning problems such as learning of halfspaces, polynomial thresholdfunctions, and regression.