Original: The Franz–Parisi Criterion and Computational Trade-offs in High Dimensional Statistics
Revised: The Franz–Parisi Criterion and Computational Trade-offs in High Dimensional Statistics

Original: Abstract
Revised: Abstract

Original: Many high-dimensional statistical inference problems are believed to possessinherent computational hardness.
Revised: Many high-dimensional statistical inference problems are believed to possess inherent computational hardness.

Original: Various frameworks have been proposed to giverigorous evidence for such hardness, including lower bounds against restrictedmodels of computation (such as low-degree functions), as well as methods rootedin statistical physics that are based on free energy landscapes.
Revised: Various frameworks have been proposed to give rigorous evidence for such hardness, including lower bounds against restricted models of computation (such as low-degree functions), as well as methods rooted in statistical physics that are based on free energy landscapes.

Original: This paper aimsto make a rigorous connection between the seemingly different low-degree andfree-energy based approaches.
Revised: This paper aims to make a rigorous connection between the seemingly different low-degree and free-energy based approaches.

Original: We deﬁne a free-energy based criterion for hardnessand formally connect it to the well-established notion of low-degree hardness for abroad class of statistical problems, namely all Gaussian additive models and certainmodels with a sparse planted signal.
Revised: We deﬁne a free-energy based criterion for hardness and formally connect it to the well-established notion of low-degree hardness for a broad class of statistical problems, namely all Gaussian additive models and certain models with a sparse planted signal.

Original: By leveraging these rigorous connectionswe are able to: establish that for Gaussian additive models the “algebraic” notionof low-degree hardness implies failure of “geometric” local MCMC algorithms,and provide new low-degree lower bounds for sparse linear regression which seemdifﬁcult to prove directly.
Revised: By leveraging these rigorous connections we are able to: establish that for Gaussian additive models the “algebraic” notion of low-degree hardness implies failure of “geometric” local MCMC algorithms, and provide new low-degree lower bounds for sparse linear regression which seem difﬁcult to prove directly.

Original: These results provide both conceptual insights into theconnections between different notions of hardness, as well as concrete technicaltools such as new methods for proving low-degree lower bounds.
Revised: These results provide both conceptual insights into the connections between different notions of hardness, as well as concrete technical tools such as new methods for proving low-degree lower bounds.

Original:  Introduction
Revised:  Introduction

Original: Many inference problems in high dimensional statistics appear to exhibit an information-computationgap , wherein at some values of the signal-to-noise ratio, inference is information-theoreticallypossible but no (time-)efﬁcient algorithm is known.
Revised: Many inference problems in high dimensional statistics appear to exhibit an information-computation gap , wherein at some values of the signal-to-noise ratio, inference is information-theoretically possible, but no (time-)efﬁcient algorithm is known.

Original: Well-known problems that exhibit such gapsinclude sparse linear regression, sparse principal component analysis (PCA), tensor PCA, plantedclique, community detection, graph coloring and many others (we point the reader to the surveyreferences [ZK16, BPW18, RSS19, KWB19, Gam21] and references therein for many examples).
Revised: Well-known problems that exhibit such gaps include sparse linear regression, sparse principal component analysis (PCA), tensor PCA, planted clique, community detection, graph coloring and many others (we point the reader to the survey references [ZK16a, BPW18, RSS19, KWB19, Gam21] and references therein for many examples).

Original: A priori, it is unclear whether such gaps are a symptom of the inherent computational intractabilityof these problems or whether they instead reﬂect a limitation of our algorithmic ingenuity.
Revised: A priori, it is unclear whether such gaps are a symptom of the inherent computational intractability of these problems, or whether they instead reﬂect a limitation of our algorithmic ingenuity.

Original: Oneof the main goals in this ﬁeld is to provide, and understand, rigorous evidence for the existence ofan information-computation gap.
Revised: One of the main goals in this ﬁeld is to provide, and understand, rigorous evidence for the existence of an information-computation gap.

Original: Indeed, there are several mature tools to establish statistical orinformation-theoretic lower bounds , and these often sharply characterize the signal-to-noise ratioat which inference is possible.
Revised: Indeed, there are several mature tools to establish statistical or information-theoretic lower bounds , and these often sharply characterize the signal-to-noise ratio at which inference is possible.

Original: We have relatively fewer tools for establishing computational lowerbounds in statistical settings, and the study of such tools is still in its early days.
Revised: We have relatively fewer tools for establishing computational lower bounds in statistical settings, and the study of such tools is still in its early days.

Original: Broadly, there arethree approaches: (i) establishing computational equivalence between suspected-to-be-hard problemsvia reductions, (ii) proving lower bounds within restricted models of computation, or in other words,ruling out families of known algorithms, and (iii) characterizing geometric properties of the problemthat tend to correspond to computational hardness, often by studying a corresponding energy orfree energy landscape of the posterior distribution of the signal given the data, and establishing the existence of ‘large barriers’ in this landscape.
Revised: Broadly, there are three approaches: (i) establishing computational equivalence between suspected-to-be-hard problems via reductions, (ii) proving lower bounds within restricted models of computation, or in other words, ruling out families of known algorithms, and (iii) characterizing geometric properties of the problem that tend to correspond to computational hardness, often by studying a corresponding energy or free energy landscape of the posterior distribution of the signal given the data, and establishing the existence of ‘large barriers’ in this landscape.

Original: In some cases it can be rigorously shown that theseproperties impede the success of certain families of algorithms (notable examples include the work of
Revised: In some cases it can be rigorously shown that these properties impede the success of certain families of algorithms (notable examples include the work of

Original: Jerrum [Jer92] and Gamarnik and Sudan [GS17]; see also references therein for other instances).
Revised: Jerrum [Jer92] and Gamarnik and Sudan [GS17]; see also references therein for other instances).

Original: These complementary approaches give us a richer understanding of the computational landscapeof high-dimensional statistical inference.
Revised: These complementary approaches give us a richer understanding of the computational landscape of high-dimensional statistical inference.

Original: Reductions contribute to establishing equivalence classesof (conjectured hard) problems, and lower bounds against restricted models, or characterizationsof the problem geometry, give concrete evidence for computational hardness within the currentlimits of known algorithms.
Revised: Reductions contribute to establishing equivalence classes of (conjectured hard) problems, and lower bounds against restricted models, or characterizations of the problem geometry, give concrete evidence for computational hardness within the current limits of known algorithms.

Original: There have been considerable recent advances in this context (see forexample [BB20], the surveys [ZK16, BPW18, RSS19, KWB19, Gam21], and references therein).
Revised: There have been considerable recent advances in this context (see for example [BB20], the surveys [ZK16a, BPW18, RSS19, KWB19, Gam21], and references therein).

Original: One particularly exciting direction, which is the topic of this paper, is the pursuit of rigorousconnections between different computational lower bound approaches.
Revised: One particularly exciting direction, which is the topic of this paper, is the pursuit of rigorous connections between different computational lower bound approaches.

Original: For instance, a recent resultshows (under mild assumptions) that lower bounds against statistical query algorithms and lowdegree polynomials are essentially equivalent [BBH + 20].
Revised: For instance, a recent result shows (under mild assumptions) that lower bounds against statistical query algorithms and low-degree polynomials are essentially equivalent [BBH + 20].

Original: Results of this type help to unify ourunderstanding about what makes problems hard.
Revised: Results of this type help to unify our understanding about what makes problems hard.

Original: Following the work of [BHK + 19, HKP + 17, HS17]in the context of the sum-of-squares hierarchy of algorithms, a conjecture was made that there is alarge and easy-to-characterize universality class of intractable problems [Hop18]: those for whichlow-degree statistics cannot distinguish data with a planted signal from (suitably deﬁned) randomnoise.
Revised: Following the work of [BHK + 19, HKP + 17, HS17] in the context of the sum-of-squares hierarchy of algorithms, a conjecture was made that there is a large and easy-to-characterize universality class of intractable problems [Hop18]: those for which low-degree statistics cannot distinguish data with a planted signal from (suitably deﬁned) random noise.

Original: These problems are “hard for the low-degree likelihood ratio” or “low-degree hard,” whichwe will deﬁne precisely below.
Revised: These problems are “hard for the low-degree likelihood ratio” or “low-degree hard,” which we will deﬁne precisely below.

Original: Many problems mentioned above fall into this class precisely in theregime of their information-computation gaps.
Revised: Many problems mentioned above fall into this class precisely in the regime of their information-computation gaps.

Original: Another successful approach to understand computational hardness of statistical problems borrowstools from statistical physics: tools such as the cavity method and replica method can be used to makeremarkably precise predictions of both statistical and computational thresholds, essentially by studyingproperties of free energy potentials associated to the problem in question, or by studying relatediterative algorithms such as belief propagation or approximate message passing (see e.g. [DKMZ11,
Revised: Another successful approach to understand computational hardness of statistical problems borrows tools from statistical physics: tools such as the cavity method and replica method can be used to make remarkably precise predictions of both statistical and computational thresholds, essentially by studying properties of free energy potentials associated to the problem in question, or by studying related iterative algorithms such as belief propagation or approximate message passing (see e.g. [DKMZ11,

Original: LKZ15a , LKZ15b, DMK + 16]) – the type of “free energy barrier” encountered in these potentialsthat is (heuristically) suggested to lead to computational hardness also often has an appealinginterpretation as a barrier to algorithms based on Markov chain Monte Carlo methods.
Revised: LKZ15a, LKZ15b, DMK + 16]) – the type of “free energy barrier” encountered in these potentials that is (heuristically) suggested to lead to computational hardness also often has an appealing interpretation as a barrier to algorithms based on Markov chain Monte Carlo methods.

Original: However, thephysics approaches are not without drawbacks, chief among them (for us) that it is notoriously difﬁcultto pin down with mathematical rigor just what the free energy potentials (certain one-dimensionalcurves) in question are, or precisely what properties of them are meant to predict computationalhardness.
Revised: However, the physics approaches are not without drawbacks, chief among them (for us) that it is notoriously difﬁcult to pin down with mathematical rigor just what the free energy potentials (certain one-dimensional curves) in question are, or precisely what properties of them are meant to predict computational hardness.

Original: Furthermore, it is clear that some versions of the “physics recipe” can make erroneouspredictions (for instance, in the case of the tensor PCA problem), and can require seemingly problemspeciﬁc modiﬁcations to be redeemed [RM14, LML + 17, BGJ20, WEM19, BCRT20, GZ19].
Revised: Furthermore, it is clear that some versions of the “physics recipe” can make erroneous predictions (for instance, in the case of the tensor PCA problem), and can require seemingly problemspeciﬁc modiﬁcations to be redeemed [RM14, LML + 17, BGJ20, WEM19, BCRT20, GZ19].

Original: Thishas made it challenging to push any mathematical account of the physics approach to predictingcomputational hardness beyond a one-problem-at-a-time theory.
Revised: This has made it challenging to push any mathematical account of the physics approach to predicting computational hardness beyond a one-problem-at-a-time theory.

Original: Main Contributions A tantalizing question and a step towards a rigorous free-energy based theoryof statistical hardness is whether some free-energy based criterion is actually rigorously connectedwith low-degree hardness.
Revised: Main Contributions A tantalizing question and a step towards a rigorous free-energy based theory of statistical hardness is whether some free-energy based criterion is actually rigorously connected with low-degree hardness.

Original: This paper aims to achieve exactly this: to make a rigorous connectionbetween the low-degree and free-energy based approaches.
Revised: This paper aims to achieve exactly this: to make a rigorous connection between the low-degree and free-energy based approaches in the setting of statistical inference.

Revised: (We note that this setting differs from that of random optimization problems with no planted signal, where a connection of this nature has already been established [GJW20, Wei22, BH21].)

Original: Our ﬁrst contribution is a formal deﬁnitionof a free-energy based criterion for computational hardness, the Franz–Parisi criterion (Deﬁnition 3.2)inspired by the so-called Franz–Parisi potential [FP95].
Revised: Our ﬁrst contribution is a formal deﬁnition of a free-energy based criterion for computational hardness, the Franz–Parisi criterion (Deﬁnition 3.2) inspired by the so-called Franz–Parisi potential [FP95].

Original: We formally connect this criterion to lowdegree hardness for a broad class of statistical problems, namely all Gaussian additive models(Theorems 4.3 and 4.4) and certain sparse planted models (e.g. Theorem 6.2).
Revised: We formally connect this criterion to low-degree hardness for a broad class of statistical problems, namely all Gaussian additive models (Theorems 4.3 and 4.4) and certain sparse planted models (e.g. Theorem 6.2).

Original: By leveraging theserigorous connections we are able to (i) establish that in the context of Gaussian additive models,low-degree hardness implies failure of local MCMC algorithms (Corollary 5.4), and (ii) providenew low-degree lower bounds for sparse linear regression which seem difﬁcult to prove directly(Theorem 6.2).
Revised: By leveraging these rigorous connections we are able to (i) establish that in the context of Gaussian additive models, low-degree hardness implies failure of local MCMC algorithms (Corollary 5.4), and (ii) provide new low-degree lower bounds for sparse linear regression which seem difﬁcult to prove directly (Theorem 6.2).

Original: In the supplementary material, we also investigate whether the Franz–Parisi criterion,as we deﬁned it, is accurate beyond Gaussian Additive Models, and ﬁnd, with counterexamples, thatit can make erroneous predictions in simple discrete settings.
Revised: In the supplementary material, we also investigate whether the Franz–Parisi criterion, as we deﬁned it, is accurate beyond Gaussian Additive Models, and ﬁnd, with counterexamples, that it can make erroneous predictions in simple discrete settings.

Original: We thus leave as an exciting futuredirection the problem of determining its precisely domain of applicability, and investigating whatother free-energy based criteria may be more suitable in other inference problems.
Revised: We thus leave as an exciting future direction the problem of determining its precise domain of applicability, and investigating what other free-energy based criteria may be more suitable in other inference problems.

Original:  Preliminaries
Revised:  Preliminaries

Original: We will focus on problems in which there is a signal vector of interest u ∈ R n , drawn from a priordistribution µ over such signals, and the data observed is a sample from a distribution P u on R N that depends on the signal u .
Revised: We will focus on problems in which there is a signal vector of interest u ∈ R n , drawn from a prior distribution µ over such signals, and the data observed is a sample from a distribution P u on R N that depends on the signal u .

Original: We focus mainly on hypothesis testing : we are given a sample generatedeither from the “planted” distribution P =
Revised: We focus mainly on hypothesis testing : we are given a sample generated either from the “planted” distribution P =

Original: E u ∼ µ P u (a mixture model were the data is drawn from P ufor a random u ∼ µ ) or from a “null” reference distribution Q representing pure noise, and the goal isto decide whether it is more likely that the sample came from P or Q .
Revised: E u ∼ µ P u (a mixture model were the data is drawn from P u for a random u ∼ µ ) or from a “null” reference distribution Q representing pure noise, and the goal is to decide whether it is more likely that the sample came from P or Q .

Original: While not our main focus, wenote that in the section on MCMC hardness we do also discuss the related task of estimation/recovery :given a sample from P u with the promise that u ∼ µ , the goal is to estimate u (different estimationerror targets correspond to different versions of this problem, often referred to weak/approximaterecovery or exact recovery ).
Revised: While not our main focus, we note that in the section on MCMC hardness we do also discuss the related task of estimation/recovery : given a sample from P u with the promise that u ∼ µ , the goal is to estimate u (different estimation error targets correspond to different versions of this problem, often referred to weak/approximate recovery or exact recovery ).

Original: We now begin with the formal deﬁnitions.
Revised: We now begin with the formal deﬁnitions.

Original: Problem 2.1 (High Dimensional Inference: Hypothesis Testing) .
Revised: Problem 2.1 (High Dimensional Inference: Hypothesis Testing) .

Original: Given positive integers n, N , anddistributions µ on R n , and P u on R N for each u ∈ supp( µ ) , the goal is to perform simple hypothesistesting between the Null model H 0 :
Revised: Given positive integers n, N , and distributions µ on R n , and P u on R N for each u ∈ supp( µ ) , the goal is to perform simple hypothesis testing between the Null model H 0 :

Original: Y ∼ Q and the Planted model H 1 : Y ∼ P =
Revised: Y ∼ Q and the Planted model H 1 : Y ∼ P =

Original: E u ∼ µ P u .
Revised: E u ∼ µ P u .

Original: We will be especially interested in asymptotic settings where n → ∞ and the other parameters scalewith n in some prescribed way: N = N n , µ = µ n , P = P n , Q = Q n .
Revised: We will be especially interested in asymptotic settings where n → ∞ and the other parameters scale with n in some prescribed way: N = N n , µ = µ n , P = P n , Q = Q n .

Original: In this setting, we focus onthe following two objectives.
Revised: In this setting, we focus on the following two objectives.

Original: Deﬁnition 2.2 (Strong/Weak Detection) .
Revised: Deﬁnition 2.2 (Strong/Weak Detection) .

Original: Strong detection is achieved if the sum of type I and type IIerrors 1 tends to 0 as n → ∞ .
Revised: Strong detection is achieved if the sum of type I and type II errors 1 tends to 0 as n → ∞ .

Original: Weak detection is achieved if the sum of type I and type II errors is atmost 1 − ε for some ﬁxed ε > 0 (not depending on n ).
Revised: Weak detection is achieved if the sum of type I and type II errors is at most 1 − ε for some ﬁxed ε > 0 (not depending on n ).

Original: Throughout, we will work in the Hilbert space L 2 ( Q ) of (square integrable) functions R N → Rwith inner product (cid:104) f, g (cid:105) Q :=
Revised: Throughout, we will work in the Hilbert space L 2 ( Q ) of (square integrable) functions R N → R with inner product (cid:104) f, g (cid:105) Q :=

Original: E Y ∼ Q [ f ( Y ) g ( Y )] and corresponding norm (cid:107) f (cid:107) Q :=
Revised: E Y ∼ Q [ f ( Y ) g ( Y )] and corresponding norm (cid:107) f (cid:107) Q :=

Original: (cid:104) f, f (cid:105) 1 / 2 Q .
Revised: (cid:104) f, f (cid:105) 1 / 2 Q .

Original: Fora function f : R N → R and integer D ∈ N , we let f ≤ D denote the orthogonal (w.r.t.
Revised: For a function f : R N → R and integer D ∈ N , we let f ≤ D denote the orthogonal (w.r.t.

Original: (cid:104)· , ·(cid:105) Q )projection of f onto the subspace of polynomials of degree ≤ D .
Revised: (cid:104)· , ·(cid:105) Q ) projection of f onto the subspace of polynomials of degree ≤ D .

Original: We assume that P u is absolutelycontinuous w.r.t.
Revised: We assume that P u is absolutely continuous w.r.t.

Original: Q for all u ∈ supp( µ ) , use L u :=
Revised: Q for all u ∈ supp( µ ) , use L u :=

Original: d P u d Q to denote the likelihood ratio, and assumethat L u ∈ L 2 ( Q ) for all u ∈ supp( µ ) .
Revised: d P u d Q to denote the likelihood ratio, and assume that L u ∈ L 2 ( Q ) for all u ∈ supp( µ ) .

Original: L := d P d Q = E u ∼ µ L u is the likelihood ratio of P and Q .
Revised: L := d P d Q = E u ∼ µ L u is the likelihood ratio of P and Q .

Original: 2.1 (Low-degree) Likelihood Ratio and Computational Complexity of Inference
Revised: 2.1 (Low-degree) Likelihood Ratio and Computational Complexity of Inference

Original: A key quantity of interest is the (squared) norm of the likelihood ratio, which is related to the chisquared divergence χ 2 ( P (cid:107) Q ) as (cid:107) L (cid:107) 2 Q = χ 2 ( P (cid:107) Q ) + 1 .
Revised: A key quantity of interest is the (squared) norm of the likelihood ratio, which is related to the chisquared divergence χ 2 ( P (cid:107) Q ) as (cid:107) L (cid:107) 2 Q = χ 2 ( P (cid:107) Q ) + 1 .

Original: If this quantity is asymptotically bounded(by O (1) or 1 + o (1) ), there are well-known consequences for information-theoretic impossibilityof testing (impossibility of strong and weak detection, respectively); see [MRZ15, Lemma 2].
Revised: If this quantity is asymptotically bounded (by O (1) or 1 + o (1) ), there are well-known consequences for information-theoretic impossibility of testing (impossibility of strong and weak detection, respectively); see [MRZ15, Lemma 2].

Original: Thelow-degree likelihood ratio , L ≤ D , which recall means the projection of the likelihood ratio ontothe subspace of degree- D polynomials, has been studied intensively in recent years as a meansto “predict” computationalcomputational complexity of high-dimensional testing problems (see[Hop18, KWB19]).
Revised: The low-degree likelihood ratio , L ≤ D , which recall means the projection of the likelihood ratio onto the subspace of degree- D polynomials, has been studied intensively in recent years as a means to “predict” computational complexity of high-dimensional testing problems (see [Hop18, KWB19]).

Original: Deﬁnition 2.3 (Low-Degree Likelihood Ratio) .
Revised: Deﬁnition 2.3 (Low-Degree Likelihood Ratio) .

Original: Deﬁne the squared norm of the degree- D likelihoodratio (also called the “low-degree likelihood ratio”) to be the quantity
Revised: Deﬁne the squared norm of the degree- D likelihood ratio (also called the “low-degree likelihood ratio”) to be the quantity

Original: [Equation]
Revised: [Equation]

Original: where the last equality follows from linearity of the projection operator, and where u, v are drawnindependently from µ .
Revised: where the last equality follows from linearity of the projection operator, and where u, v are drawn independently from µ .

Original: For some increasing sequence D = D n , we say that the hypothesis testingproblem above is hard for the degree- D likelihood or simply low-degree hard if LD( D )
Revised: For some increasing sequence D = D n , we say that the hypothesis testing problem above is hard for the degree- D likelihood or simply low-degree hard if LD( D )

Original: =
Revised: =

Original: O (1) .
Revised: O (1) .

Original: Heuristically speaking, the interpretation of LD( D ) should be thought of as analogous to that of (cid:107) L (cid:107) 2 Qbut for computationally-bounded tests: if LD( D ) =
Revised: Heuristically speaking, the interpretation of LD( D ) should be thought of as analogous to that of (cid:107) L (cid:107) 2 Q but for computationally-bounded tests: if LD( D ) =

Original: O (1) (or 1 + o (1) )
Revised: O (1) (or 1 + o (1) )

Original: this suggests computationalhardness of strong (or weak, respectively) detection.
Revised: this suggests computational hardness of strong (or weak, respectively) detection.

Original: The parameter D
Revised: The parameter D

Original: = D n should be looselythought of as a proxy for the runtime allowed for our testing algorithm, where D =
Revised: = D n should be loosely thought of as a proxy for the runtime allowed for our testing algorithm, where D =

Original: O (log n )corresponds to polynomial time and more generally, larger values of D correspond to runtimeexp(˜Θ( D )) where ˜Θ hides factors of log n .
Revised: O (log n ) corresponds to polynomial time and more generally, larger values of D correspond to runtime exp(˜Θ ( D )) where ˜Θ hides factors of log n .

Original: See supplementary material for further discussion of therelationship between low-degree hardness and hardness against other classes of algorithms.
Revised: See supplementary material for further discussion of the relationship between low-degree hardness and hardness against other classes of algorithms.

Original:  The Franz Parisi criterion
Revised:  The Franz Parisi criterion

Original: We deﬁne a predictor of computational hardness, which we call the Franz–Parisi criterion .
Revised: We deﬁne a predictor of computational hardness, which we call the Franz–Parisi criterion .

Original: It isinspired by well-established ideas rooted in statistical physics, which we discuss further below andin the supplement.
Revised: It is inspired by well-established ideas rooted in statistical physics, which we discuss further below and in the supplement.

Original: However, the precise deﬁnition we use here has not appeared before (to ourknowledge).
Revised: However, the precise deﬁnition we use here has not appeared before (to our knowledge).

Original: Throughout this paper we will argue for the signiﬁcance of this criterion in a number ofways: its conceptual link to physics (Remark 3.3), its provable equivalence to the low-degree criterionfor Gaussian additive models, its formal connection to MCMC methods for Gaussian additive models(Section 5), and its usefulness as a tool for proving low-degree lower bounds (Section 6).
Revised: Throughout this paper we will argue for the signiﬁcance of this criterion in a number of ways: its conceptual link to physics, its provable equivalence to the low-degree criterion for Gaussian additive models, its formal connection to MCMC methods for Gaussian additive models (Section 5), and its usefulness as a tool for proving low-degree lower bounds (Section 6).

Original: Deﬁnition 3.1 (Low-Overlap Likelihood Norm) .
Revised: Deﬁnition 3.1 (Low-Overlap Likelihood Norm) .

Original: We deﬁne the low-overlap likelihood norm atoverlap δ ≥ 0 as
Revised: We deﬁne the low-overlap likelihood norm at overlap δ ≥ 0 as

Original: [Equation]
Revised: [Equation]

Original: where u, v are drawn independently from µ .
Revised: where u, v are drawn independently from µ .

Original: Deﬁnition 3.2 (Franz–Parisi Criterion) .
Revised: Deﬁnition 3.2 (Franz–Parisi Criterion) .

Original: We deﬁne the Franz–Parisi Criterion at D deviations to bethe quantityFP [Equation] LO
Revised: We deﬁne the Franz–Parisi Criterion at D deviations to be the quantity

Original: [Equation]
Revised: [Equation]

Original: for [Equation] [Equation] For some increasing sequence D = D n , a problem is FP-hard at D deviations if FP( D ) =
Revised: For some increasing sequence D = D n , a problem is FP-hard at D deviations if FP( D ) =

Original: O (1) .
Revised: O (1) .

Original: Heuristically speaking, FP( D ) should be thought of as having a similar interpretation as LD( D ) : if
Revised: Heuristically speaking, FP( D ) should be thought of as having a similar interpretation as LD( D ) : if

Original: FP( D ) =
Revised: FP( D ) =

Original: O (1) this suggests hardness of strong detection, and if FP( D )
Revised: O (1) this suggests hardness of strong detection, and if FP( D )

Original: =
Revised: =

Original: 1 + o (1) this suggestshardness of weak detection.
Revised: 1 + o (1) this suggests hardness of weak detection.

Original: The parameter D is a proxy for runtime and corresponds to the parameter
Revised: The parameter D is a proxy for runtime and corresponds to the parameter

Original: D in LD( D ) , as we justify below.
Revised: D in LD( D ) , as we justify below.

Original: We remark that LD( D ) and FP( D ) can be thought of as different ways of “restricting” the quantity
Revised: We remark that LD( D ) and FP( D ) can be thought of as different ways of “restricting” the quantity

Original: [Equation]
Revised: [Equation]

Original: which, recall, is related to information-theoretic impossibility of testing.
Revised: which, recall, is related to information-theoretic impossibility of testing.

Original: For LD , the restriction islow-degree projection on each L u , while for FP it excludes pairs ( u, v ) of high overlap.
Revised: For LD , the restriction is low-degree projection on each L u , while for FP it excludes pairs ( u, v ) of high overlap.

Original: Our resultswill show that (in some settings) these two restrictions are nearly equivalent.
Revised: Our results will show that (in some settings) these two restrictions are nearly equivalent.

Original: Remark 3.3 (Relation to statistical physics) . We defer to supplementary material an extendeddiscussion of the relationship between the Franz–Parisi criterion and ideas from statistical physics. 
Revised: Relation to statistical physics We defer to supplementary material an extended discussion of the relationship between the Franz–Parisi criterion and ideas from statistical physics. See also [MM09,

Revised: ZK16b] for exposition on the well-explored connections between statistical physics and Bayesian inference.

Original: Roughly speaking, the logarithm of LO( δ ) is the ‘annealed’ approximation near δ = 0 of the Franz–
Revised: Roughly speaking, the logarithm of LO( δ ) is the ‘annealed’ approximation near δ =of the Franz–

Original: Parisi potential [FP95]; a tool used to study the free energy landscape of a disordered system locallyaround a reference conﬁguration at equilibrium.
Revised: Parisi potential [FP95]; a tool used to study the free energy landscape of a disordered system locally around a reference conﬁguration at equilibrium.

Original: In our statistical context this potential takes the form
Revised: In our statistical context this potential takes the form

Original: [Equation]
Revised: [Equation]

Revised: Roughly speaking, f ( δ ) measures the amount of mass that the posterior distribution places on vectors of overlap δ with the true signal.

Original: Jensen’s inequality leads to the upper bound f ( 
Revised: Jensen’s inequality leads to the upper bound f ( δ ) ≤ f ann ( δ ) :

Original: [Equation]
Revised: = log E u,v ∼ µ (cid:2) 1 (cid:104) u,v (cid:105) = δ (cid:104)

Original: ,which we call the annealed FP potential , c.f. LO( δ ) , Eq.
Revised: L u , L v (cid:105) Q (cid:3) , which we call the annealed FP potential , c.f. LO( δ ) , Eq.

Original: For inference problems which are information-theoretically solvable, most of the mass of the posteriordistribution of v | Y , when Y ∼ P u is concentrated on those v with large (cid:104) u, v (cid:105) , and hence theseare the main contributors to  the partition function.
Revised: For inference problems which are information-theoretically solvable, most of the mass of the posterior distribution of v | Y , when Y ∼ P u is concentrated on those v with large (cid:104) u, v (cid:105) . But to predict computational complexity, the Franz–Parisi criterion stipulates that it is the contributions of v with |(cid:104) u, v (cid:105)| ≤ δ for small values of δ which matter: when such v contribute noticeably, LO( δ ) is large and hence so is FP( D ) .

Original: But topredict computational complexity, the 
Revised: The FP criterion, Deﬁnition 3.2, is whether FP( D ) stays bounded or diverges as n → ∞ for some choice of an increasing sequence D = D n . A heuristic justiﬁcation of this criterion is as follows: We should ﬁrst note that since L u ≥ 0 , we have FP( D ) ≤ (cid:107) L (cid:107) 2 Q . Thus if (cid:107) L (cid:107) Q → ∞ but FP( D ) = O (1) , then the divergence of (cid:107) L (cid:107) Q must be due to contributions to the sum Eq.

Original: Franz–Parisi criterion stipulates that it is the contributions of v with  |(cid:104) u, v (cid:105)| ≤ δ for small values of δwhich matter: when such v contribute noticeably, LO( δ  ) is large and hence so is FP( D ) . 
Revised: (4) with high overlap values: |(cid:104) u, v (cid:105)| (cid:29) δ ( D ) . Suppose now there is a free energy barrier separating small overlaps |(cid:104) u, v (cid:105)| ≤ δ ( D ) from larger ones |(cid:104) u, v (cid:105)| (cid:29) δ ( D ) . For instance, suppose (cid:104) u, v (cid:105) =is a local maximum of the potential, separated by a barrier from a global maximum located at (cid:104) u, v (cid:105) (cid:29) δ ( D ) (see Fig.

Revised: 1, Panel ( b ) ), then one needs much more than e O ( D ) samples to guess an overlap value on the other side of the barrier and land in the basin of attraction of the global maximum.

Revised: This suggests that tests distinguishing P and Q cannot be constructed in time e O ( D ) .

Revised: One of our main results (see Section 4) is an equivalence relation between the FP( D ) criterion and the LD( D (cid:48) ) criterion for Gaussian models, where D (cid:48)

Original:  Example: The spiked Wigner model As a concrete example, let us consider the spiked Wignermodel with sparse Rademacher prior: The signal vector u has i.i.d.
Revised: = ˜Θ( D ) , therefore grounding this heuristic in a rigorous statement. Example: The spiked Wigner model As a concrete example, let us consider the spiked Wigner model with sparse Rademacher prior: The signal vector u has i.i.d.

Original: entries drawn from a three-pointprior µ 0 = ρ 2 δ +1 / √ ρ
Revised: entries drawn from a three-point prior µ 0 = ρ 2 δ +1 / √ ρ

Original: +(1 − ρ )
Revised: +(1 − ρ )

Original: δ 0 + ρ 2 δ − 1 / √ ρ , and for 1 ≤ i ≤ j ≤ n we let Y ij = λ √ n u i u j + Z ij , where
Revised: δ 0 + ρ 2 δ − 1 / √ ρ , and for 1 ≤ i ≤ j ≤ n we let Y ij = λ √ n u i u j + Z ij , where

Original: Z is drawn from the Gaussian Orthogonal Ensemble: Z ij ∼ N (0 , 1) for i < j and Z ii ∼ N (0 , 2) .
Revised: Z is drawn from the Gaussian Orthogonal Ensemble: Z ij ∼ N (0 , 1) for i < j and Z ii ∼ N (0 , 2) .

Original: The null distribution is pure Gaussian noise: Y =
Revised: The null distribution is pure Gaussian noise: Y =

Original: Z .
Revised: Z .

Original: In this case, known efﬁcient algorithms succeedat detecting/estimating the signal u if and only if λ > 1 [BGN11, LM19, CL19].
Revised: In this case, known efﬁcient algorithms succeed at detecting/estimating the signal u if and only if λ > 1 [BGN11, LM19, CL19].

Original: Furthermore, thethreshold λ ALG = 1 is information-theoretically tight when ρ = 1 (or more generally, if ρ is largerthan a known absolute constant).
Revised: Furthermore, the threshold λ ALG = 1 is information-theoretically tight when ρ = 1 (or more generally, if ρ is larger than a known absolute constant).

Original: On the other hand if ρ is small enough, then detection becomesinformation-theoretically possible for some λ < 1 but no known polynomial-time algorithm succeeds [Figure]{+
+}
Revised: On the other hand if ρ is small enough, then detection becomes information-theoretically possible for some λ < {+
+}

Original: A nn e a l e d F P [Figure] in the regime [BMV + 17, PWBM18, AKJ20].
Revised: but no known polynomial-time algorithm succeeds in the regime [BMV + 17, PWBM18, AKJ20].

Original: Let us check that the behavior of the annealed potentialis qualitatively consistent with these facts.
Revised: Let us check that the behavior of the annealed potential is qualitatively consistent with these facts.

Original: A small computation (see Section 4) leads to the expression
Revised: A small computation (see Section 4) leads to the expression

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: and the annealed FP potential is
Revised: and the annealed FP potential is

Original: [Equation]
Revised: [Equation]

Original: Letting k
Revised: Letting k

Original: = (cid:98) nx
Revised: = (cid:98) nx

Original: (cid:99) , and using Stirling’s formula, we obtain a variational formula for the annealed FPpotential: f ann ( (cid:98) nx (cid:99) /ρ )
Revised: (cid:99) , and using Stirling’s formula, we obtain a variational formula for the annealed FP potential: f ann ( (cid:98) nx (cid:99) /ρ )

Original: =
Revised: =

Original: nφ ( x ) + o ( n ) where
Revised: nφ ( x ) + o ( n ) where

Original: [Equation]
Revised: [Equation]

Original: The max is over probability vectors p
Revised: The max is over probability vectors p

Original: = ( p − 1 , p 0 , p 1 ) satisfying p 1 − p − 1 = x , and h ( p ) =
Revised: = ( p − 1 , p 0 , p 1 ) satisfying p 1 − p − 1 = x , and h ( p ) =

Original: − p − 1 log p − 1 − p 0 log p 0 − p 1 log p 1 .
Revised: − p − 1 log p − 1 − p 0 log p 0 − p 1 log p 1 .

Original: It is not difﬁcult to check that φ (0) =
Revised: It is not difﬁcult to check that φ (0) =

Original: φ (cid:48) (0) = 0 , andφ (cid:48)(cid:48) (0) =
Revised: φ (cid:48) (0) = 0 , and φ (cid:48)(cid:48) (0) =

Original: ( λ 2 − 1) /ρ 2 .
Revised: ( λ 2 − 1) /ρ 2 .

Original: Hence when λ < λ ALG = 1 , f ann is negative for all | x | ≤ ε for someε =
Revised: Hence when λ < λ ALG = 1 , f ann is negative for all | x | ≤ ε for some ε

Original:  ε ( λ, ρ ) (Fig 1, Panels ( a ) , ( b ) ).
Revised: = ε ( λ, ρ ) (Fig 1, Panels ( a ) , ( b ) ).

Original: This indicates that FP( D ) is bounded for D ≤ c ( ε ) n .
Revised: This indicates that FP( D ) is bounded for D ≤ c ( ε ) n .

Original: On theother hand, if λ > 1 , f ann > 0 over [ − ε, ε ] for some ε > 0 , which indicates that FP( D ) → ∞ for
Revised: On the other hand, if λ > 1 , f ann > 0 over [ − ε, ε ] for some ε > 0 , which indicates that FP( D ) → ∞ for

Original: D = D n → ∞ slowly (Panel ( c ) ).
Revised: D = D n → ∞ slowly (Panel ( c ) ).

Original: One can also look at the global behavior of f ann , which we plot in
Revised: One can also look at the global behavior of f ann , which we plot in

Original: Figure 1.
Revised: Figure 1.

Original: Panel ( b ) represents a scenario where x = 0 is a local minimum separated from the twosymmetric global maxima by a barrier, while in panels ( a ) and ( c ) , no such barrier exists.
Revised: Panel ( b ) represents a scenario where x = 0 is a local maximum separated from the two symmetric global maxima by a barrier, while in panels ( a ) and ( c ) , no such barrier exists.

Original: Differences to prior physical hardness criteria Our FP criterion is conceptually similar to ideasthat have appeared before in statistical physics, but there are a few key differences.
Revised: Differences to prior physical hardness criteria Our FP criterion is conceptually similar to ideas that have appeared before in statistical physics, but there are a few key differences.

Original: While free energybarriers are typically thought of as an obstruction to algorithmic recovery , our criterion—due to itsconnection with (cid:107) L (cid:107) 2 Q —is instead designed for the detection problem.
Revised: While free energy barriers are typically thought of as an obstruction to algorithmic recovery , our criterion—due to its connection with (cid:107) L (cid:107) 2 Q —is instead designed for the detection problem.

Original: As such, we do not expect theannealed FP to make sharp predictions about estimation error (e.g. MMSE) like approaches based onapproximate message passing (AMP), e.g. [DMM09, LKZ15a, LKZ15b, DMK + 16].
Revised: As such, we do not expect the annealed FP to make sharp predictions about estimation error (e.g. MMSE) like approaches based on approximate message passing (AMP), e.g. [DMM09, LKZ15a, LKZ15b, DMK + 16].

Original: (For instance,
Revised: (For instance,

Revised: [Figure]

Original:  Figure 1, Panel ( b ) wrongly predicts that estimation is ‘possible but hard’ for λ = 0 .
Revised: A nn e a l e d F P [Figure] Figure 1, Panel ( b ) wrongly predicts that estimation is ‘possible but hard’ for λ = 0 .

Original: 9 , ρ = 0 .
Revised: 9 , ρ = 0 .

Original: 4 , whenit is in fact information-theoretically impossible [LM19].)
Revised: 4 , when it is in fact information-theoretically impossible [LM19].)

Original: On the other hand, one advantage of ourcriterion is that, by virtue of its connection to LD in Section 4, it predicts the correct computationalthreshold 2 for tensor PCA (matching the best known poly-time algorithms), whereas existing methodsbased on AMP or free energy barriers capture a different threshold [RM14, LML + 17, BGJ20] (seealso [WEM19, BCRT20] for other ways to “redeem” the physics approach).
Revised: On the other hand, one advantage of our criterion is that, by virtue of its connection to LD in Section 4, it predicts the correct computational threshold 2 for tensor PCA (matching the best known poly-time algorithms), whereas existing methods based on AMP or free energy barriers capture a different threshold [RM14, LML + 17 , BGJ20] (see also [WEM19, BCRT20] for other ways to “redeem” the physics approach).

Original: One-sidedness of hardness criteria Finally, we note that the quantities (cid:107) L (cid:107) 2 Q , LD( D ) , FP( D )should be thought of primarily as lower bounds that imply/suggest impossibility or hardness.
Revised: One-sidedness of hardness criteria Finally, we note that the quantities (cid:107) L (cid:107) 2 Q , LD( D ) , FP( D ) should be thought of primarily as lower bounds that imply/suggest impossibility or hardness.

Original: If oneof these quantities does not remain bounded as n → ∞ , it does not necessarily mean the problemis possible/tractable.
Revised: If one of these quantities does not remain bounded as n → ∞ , it does not necessarily mean the problem is possible/tractable.

Original: We will revisit this issue again in Section 6, where a conditional low-degreecalculation will be used to prove hardness even though the standard LD blows up (akin to theconditional versions of (cid:107) L (cid:107) 2 Q that are commonly used to prove information-theoretic lower bounds,e.g. [BMNN16, BMV + 17, PWB16, PWBM18]).
Revised: We will revisit this issue again in Section 6, where a conditional low-degree calculation will be used to prove hardness even though the standard LD blows up (akin to the conditional versions of (cid:107) L (cid:107) 2 Q that are commonly used to prove information-theoretic lower bounds, e.g. [BMNN16, BMV + 17, PWB16, PWBM18]).

Original:  The Gaussian Additive Model
Revised:  The Gaussian Additive Model

Original: We will for now focus on a particular class of estimation models, the so called Gaussian additivemodels .
Revised: We will for now focus on a particular class of estimation models, the so called Gaussian additive models .

Original: The distribution P u describes an observation of the form
Revised: The distribution P u describes an observation of the form

Original: [Equation]
Revised: [Equation]

Original: (6) where λ ≥ 0 is the signal-to-noise ratio, u ∼ µ is the signal of interest drawn from some distributionµ on R N , and Z ∼ N (0 , I N ) is standard Gaussian noise (independent from u ).
Revised: where λ ≥ 0 is the signal-to-noise ratio, u ∼ µ is the signal of interest drawn from some distribution µ on R N , and Z ∼ N (0 , I N ) is standard Gaussian noise (independent from u ).

Original: In recovery, the goalis to recover u , or more precisely to compute an estimator ˆ u ( Y ) that correlates with u .
Revised: In recovery, the goal is to recover u , or more precisely to compute an estimator ˆ u ( Y ) that correlates with u .

Original: We note thatin principle, λ could be absorbed into the norm of u , but it will be convenient for us to keep λ explicitbecause some of our results will involve perturbing λ slightly.
Revised: We note that in principle, λ could be absorbed into the norm of u , but it will be convenient for us to keep λ explicit because some of our results will involve perturbing λ slightly.

Original: We focus on the hypothesis testingversion of this question, where the goal is to distinguish a sample (6) from a standard Gaussian vector.
Revised: We focus on the hypothesis testing version of this question, where the goal is to distinguish a sample (6) from a standard Gaussian vector.

Original: Deﬁnition 4.1 (Gaussian Additive Model: Hypothesis Testing) .
Revised: Deﬁnition 4.1 (Gaussian Additive Model: Hypothesis Testing) .

Original: Given N a positive integer, λ ≥ 0 ,and µ a distribution on R N with all moments ﬁnite, let Z ∼ N (0 , I ) and u ∼ µ ; then hypothesistesting in the Gaussian additive model consists of performing a simple hypothesis test between
Revised: Given N a positive integer, λ ≥ 0 , and µ a distribution on R N with all moments ﬁnite, let Z ∼ N (0 , I ) and u ∼ µ ; then hypothesis testing in the Gaussian additive model consists of performing a simple hypothesis test between

Original: H
Revised: H

Original: Q : Y =
Revised: [Equation]

Original: Z and
Revised: and

Original: H P : Y = λu + Z .
Revised: [Equation]

Original: We are interested in understanding, as N = N n → ∞ , for which prior distributions µ = µ n and
Revised: We are interested in understanding, as N = N n → ∞ , for which prior distributions µ = µ n and

Original: SNR levels λ = λ n it is possible to computationally efﬁciently distinguish a sample from P from asample from Q (in the sense of strong or weak detection; see Deﬁnition 2.2).
Revised: SNR levels λ = λ n it is possible to computationally efﬁciently distinguish a sample from P from a sample from Q (in the sense of strong or weak detection; see Deﬁnition 2.2).

Original: As in Section 2 we use224 P u to denote the distribution N ( λu, I ) , and write L u = d P u d Q .
Revised: As in Section 2 we use P u to denote the distribution N ( λu, I ) , and write L u = d P u d Q .

Original: A number of classical inference tasks are captured by the Gaussian additive model, and bounds on
Revised: A number of classical inference tasks are captured by the Gaussian additive model, and bounds on

Original: LD( D ) are known for many of them [HKP + 17, KWB19, DKWB19, BBK + 21], including matrixand tensor PCA.
Revised: LD( D ) are known for many of them [HKP + 17, KWB19, DKWB19, BBK + 21], including matrix and tensor PCA.

Original: Example 4.2 (Matrix and Tensor PCA) .
Revised: Example 4.2 (Matrix and Tensor PCA) .

Original: In the matrix PCA case, we take N = n 2 and u = x ⊗ 2 ,where x is for instance drawn with i.i.d.
Revised: In the matrix PCA case, we take N = n 2 and u = x ⊗ 2 , where x is for instance drawn with i.i.d.

Original: coordinates from some prior µ 0 .
Revised: coordinates from some prior µ 0 .

Original: In the tensor case, we take
Revised: In the tensor case, we take

Original: N = n p and u = x ⊗ p , with x drawn uniformly from the sphere S n − 1 .
Revised: N = n p and u = x ⊗ p , with x drawn uniformly from the sphere S n − 1 .

Original: FP-LD Equivalence
Revised: FP-LD Equivalence

Original: The following two theorems show that in the Gaussian additive model, FPand LD are equivalent up to logarithmic factors in D and 1 + ε factors in λ .
Revised: The following two theorems show that in the Gaussian additive model, FP and LD are equivalent up to logarithmic factors in D and 1 + ε factors in λ .

Original: Recall the notation
Revised: Recall the notation

Original: LD( D ) and FP( D ) from (1) and (3), here also with a dependency on the SNR λ .
Revised: LD( D ) and FP( D ) from (1) and (3), here also with a dependency on the SNR λ .

Original: Theorem 4.3 (FP-hard implies LD-hard) .
Revised: Theorem 4.3 (FP-hard implies LD-hard) .

Original: Assume the Gaussian additive model (Deﬁnition 4.1) andsuppose (cid:107) u (cid:107) 2 ≤ M for all u ∈ supp( µ ) , for some M > 0 .
Revised: Assume the Gaussian additive model (Deﬁnition 4.1) and suppose (cid:107) u (cid:107) 2 ≤ M for all u ∈ supp( µ ) , for some M > 0 .

Original: Then for any λ ≥ 0 and any odd integer
Revised: Then for any λ ≥ 0 and any odd integer

Original: D ≥ 1 , LD( D, λ ) ≤ FP( ˜ D, λ ) + e − D , where ˜ D :
Revised: D ≥ 1 , LD( D, λ ) ≤ FP( ˜ D, λ ) + e − D , where ˜ D :

Original: = D · (2 + log(1 + λ 2 M )) .
Revised: = D · (2 + log(1 + λ 2 M )) .

Original: We give the proof in the supplement.
Revised: We give the proof in the supplement.

Original: We are primarily interested in the regime D = ω (1) , where wehave shown that LD can only exceed FP by an additive o (1) term.
Revised: We are primarily interested in the regime D = ω (1) , where we have shown that LD can only exceed FP by an additive o (1) term.

Original: We have lost logarithmic factors inpassing from D to ˜ D .
Revised: We have lost logarithmic factors in passing from D to ˜ D .

Original: For many applications, these log factors are not an issue because (in the “hard”regime) FP is bounded for some D
Revised: For many applications, these log factors are not an issue because (in the “hard” regime) FP is bounded for some D

Original: = N Ω(1) while λ, M are polynomial in N .
Revised: = N Ω(1) while λ, M are polynomial in N .

Original: Theorem 4.4 (LD-hard implies FP-hard) .
Revised: Theorem 4.4 (LD-hard implies FP-hard) .

Original: Assume the Gaussian additive model (Deﬁnition 4.1).
Revised: Assume the Gaussian additive model (Deﬁnition 4.1).

Original: Forall ε ∈ (0 , 1) there exists D 

Revised: For all ε ∈ (0 , 1) there exists D

Original: = D 0 ( ε ) > 0 such that for any λ ≥ 0 and any even integer D ≥ D 0 , if
Revised: = D 0 ( ε ) > 0 such that for any λ ≥ 0 and any even integer D ≥ D 0 , if

Original: LD( D, (1 + ε ) λ ) ≤ (1 + ε ) D / ( eD ) then
Revised: LD( D, (1 + ε ) λ ) ≤ (1 + ε ) D / ( eD ) then

Original: FP( D, λ ) ≤ LD( D, (1 + ε ) λ ) + ε
Revised: FP

Original: [Equation]
Revised: [Equation]

Original: The proof, together with proofs for various corollaries, can be found in the supplementary material.
Revised: The proof, together with proofs for various corollaries, can be found in the supplementary material.

Original: In the asymptotic regime of primary interest, we have the following consequence.
Revised: In the asymptotic regime of primary interest, we have the following consequence.

Original: Corollary 4.5.
Revised: Corollary 4.5.

Original: Fix any constant ε (cid:48) > 0 and suppose D = D n , λ = λ n , N = N n , and µ = µ nare such that D is an even integer, D = ω (1) , and LD( D, (1 + ε (cid:48) )
Revised: Fix any constant ε (cid:48) > 0 and suppose D = D n , λ = λ n , N = N n , and µ = µ n are such that D is an even integer, D = ω (1) , and LD( D, (1 + ε (cid:48) )

Original: λ )
Revised: λ )

Original: =
Revised: =

Original: O (1) .
Revised: O (1) .

Original: Then FP( D, λ ) ≤
Revised: Then FP( D, λ ) ≤

Original: LD( D, (1 + ε (cid:48) )
Revised: LD( D, (1 + ε (cid:48) )

Original: λ )
Revised: λ )

Original: + o (1) .
Revised: + o (1) .

Original: Remark 4.6.
Revised: Remark 4.6.

Original: Above, we have taken the liberty to assume D has a particular parity for convenience.
Revised: Above, we have taken the liberty to assume D has a particular parity for convenience.

Original: Since FP and LD are both monotone in D , one can readily deduce similar results for all integers D .
Revised: Since FP and LD are both monotone in D , one can readily deduce similar results for all integers D .

Original:  FP-Hard Implies MCMC-Hard
Revised:  FP-Hard Implies MCMC-Hard

Original: In this section we show that in the Gaussian additive model, if FP is uniformly bounded then anatural class of local Markov chain Monte Carlo (MCMC) methods fail to recover the planted signal.
Revised: In this section we show that in the Gaussian additive model, if FP is uniformly bounded then a natural class of local Markov chain Monte Carlo (MCMC) methods fail to recover the planted signal.

Original: Combining this with the results of the previous section, we also ﬁnd that low-degree hardness, animpossibility result for polynomial (“algebraic”) methods, implies MCMC-hardness, which is thefailure of “geometric” methods.
Revised: Combining this with the results of the previous section, we also ﬁnd that low-degree hardness, an impossibility result for polynomial (“algebraic”) methods, implies MCMC-hardness, which is the failure of “geometric” methods.

Original: To the best of our knowledge this is the ﬁrst time such a directimplication is established for any  model.
Revised: To the best of our knowledge this is the ﬁrst time such a direct implication is established for any planted model.

Original: Note also that such a result is interesting as low-degreehardness is in principle an impossibility result for detection, while MCMC-hardness is a failure ofsome MCMC methods to perform estimation.
Revised: Note also that such a result is interesting as lowdegree hardness is in principle an impossibility result for detection, while MCMC-hardness is a failure of some MCMC methods to perform estimation.

Revised: We remark that the result of this section is similar in spirit to [BAJ18], which relates a version of annealed FP to the spectral gap for Langevin dynamics (albeit in a setting with no planted solution).

Original: We again restrict our attention to the additive Gaussian model, now with the additional assumptionthat the prior µ is uniform on some ﬁnite set S ⊆ R N with transitive symmetry , deﬁned as follows.
Revised: We again restrict our attention to the additive Gaussian model, now with the additional assumption that the prior µ is uniform on some ﬁnite set S ⊆ R N with transitive symmetry , deﬁned as follows.

Original: Deﬁnition 5.1.
Revised: Deﬁnition 5.1.

Original: We say
Revised: We say

Original: S ⊆ R N has transitive symmetry if for any u, v ∈ S there exists an orthogonalmatrix R ∈ O( N ) such that Ru = v and RS = S .
Revised: S ⊆ R N has transitive symmetry if for any u, v ∈ S there exists an orthogonal matrix R ∈ O( N ) such that Ru = v and RS = S .

Original: The assumption that S is ﬁnite is not too restrictive because one can approximate any continuous priorto arbitrary accuracy by a discrete set (see the supplementary material for more details on that).
Revised: The assumption that S is ﬁnite is not too restrictive because one can approximate any continuous prior to arbitrary accuracy by a discrete set (see the supplementary material for more details on that).

Original: Notethat transitive symmetry implies that every u ∈ S has the same 2-norm.
Revised: Note that transitive symmetry implies that every u ∈ S has the same 2-norm.

Original: Without loss of generalitywe will assume this 2-norm is 1, i.e., S is a subset of the unit sphere S N − 1 .
Revised: Without loss of generality we will assume this 2-norm is 1, i.e., S is a subset of the unit sphere S N − 1 .

Original: Given an observation Y = λu + Z from the Gaussian additive model (with µ uniform on a ﬁnite,transitive-symmetric set S ⊆ S N − 1 ), consider the associated Gibbs measure ν β on S deﬁned by
Revised: Given an observation Y = λu + Z from the Gaussian additive model (with µ uniform on a ﬁnite, transitive-symmetric set S ⊆ S N − 1 ), consider the associated Gibbs measure ν β on S deﬁned by

Original: [Equation]
Revised: [Equation]

Original: (8) where β ≥ 0 is an inverse-temperature parameter (i.e., 1 /β is the temperature ), H ( v ) = −(cid:104) v, Y (cid:105)is the Hamiltonian , and Z β = (cid:80) v ∈ S exp( − βH ( v )) is the partition function .
Revised: ) where β ≥ 0 is an inverse-temperature parameter (i.e., 1 /β is the temperature ), H ( v ) = −(cid:104) v, Y (cid:105) is the Hamiltonian , and Z β = (cid:80) v ∈ S exp( − βH ( v )) is the partition function .

Original: When β = λ (the“Bayesian temperature”), the Gibbs measure ν β is precisely the posterior distribution for the signal ugiven the observation 
Revised: When β = λ (the “Bayesian temperature”), the Gibbs measure ν β is precisely the posterior distribution for the signal u given the observation Y =

Original: Y = λu + Z .
Revised: λu + Z .

Original: We consider an arbitrary Markov chain X 0 , X 1 , X 2 , . . .
Revised: We consider an arbitrary Markov chain X 0 , X 1 , X 2 , . . .

Original: on state space S with stationary distributionν β (for some β ), i.e., if X t ∼ ν β then X t +1 ∼ ν β .
Revised: on state space S with stationary distribution ν β (for some β ), i.e., if X t ∼ ν β then X t +1 ∼ ν β .

Original: We will assume a worst-case initial state, whichmay depend adversarially on Y .
Revised: We will assume a worst-case initial state, which may depend adversarially on Y .

Original: We are interested in hitting time lower bounds , showing that such a
Revised: We are interested in hitting time lower bounds , showing that such a

Original: Markov chain will take many steps before ﬁnding a “good” state that is close to the true signal u .
Revised: Markov chain will take many steps before ﬁnding a “good” state that is close to the true signal u .

Original: The core idea of our argument is to establish a free energy barrier , that is, a subset B ⊆ S of small
Revised: The core idea of our argument is to establish a free energy barrier , that is, a subset B ⊆ S of small

Original: Gibbs mass that separates the initial state from the “good” states.
Revised: Gibbs mass that separates the initial state from the “good” states.

Original: Such a barrier is well-known toimply a lower bound for the hitting time of the “good” states using conductance; see e.g. [LP17,
Revised: Such a barrier is well-known to imply a lower bound for the hitting time of the “good” states using conductance; see e.g. [LP17,

Original: Theorem 7.4].
Revised: Theorem 7.4].

Original: This is formalized in the proposition below.
Revised: This is formalized in the proposition below.

Original: Proposition 5.2 (Free Energy Barrier Implies Hitting Time Lower Bound) .
Revised: Proposition 5.2 (Free Energy Barrier Implies Hitting Time Lower Bound) .

Original: Suppose X 0 , X 1 , X 2 , . . .is
Revised: Suppose X 0 , X 1 , X 2 , . . .

Original:  a Markov chain on a ﬁnite state space S , with some stationary distribution ν .
Revised: is a Markov chain on a ﬁnite state space S , with some stationary distribution ν .

Original: Let A and B be twodisjoint subsets of S and deﬁne the hitting time τ B := inf { t ∈ N : X t ∈ B } .
Revised: Let A and B be two disjoint subsets of S and deﬁne the hitting time τ B := inf { t ∈ N : X t ∈ B } .

Original: If the initial state X 0 isdrawn from the conditional distribution ν | A , then for any t ∈ N , Pr( τ B ≤ t ) ≤ t · ν ( B ) ν
Revised: If the initial state X 0 is drawn from the conditional distribution ν | A , then for any t ∈ N , Pr( τ B ≤ t ) ≤ t · ν ( B ) ν

Original: ( A ) .
Revised: ( A ) .

Original: In particular,for any t ∈ N there exists a v ∈ A s.t.
Revised: In particular, for any t ∈ N there exists a v ∈ A s.t.

Original: if X 0 = v deterministically, then Pr( τ B ≤ t ) ≤ t · ν ( B ) ν ( A ) .
Revised: if X 0 = v deterministically, then Pr( τ B ≤ t ) ≤ t · ν ( B ) ν ( A ) .

Original: We will need to impose some “locality” on our Markov chain so that it cannot jump from A to agood state without ﬁrst visiting B .
Revised: We will need to impose some “locality” on our Markov chain so that it cannot jump from A to a good state without ﬁrst visiting B .

Original: We say a Markov chain on a ﬁnite state space S ⊆ S N − 1 is∆ -local if for every possible transition v → v (cid:48)
Revised: We say a Markov chain on a ﬁnite state space S ⊆ S N − 1 is ∆ -local if for every possible transition v → v (cid:48)

Original: we have (cid:107) v − v (cid:48) (cid:107) 2 ≤ ∆ .
Revised: we have (cid:107) v − v (cid:48) (cid:107) 2 ≤ ∆ .

Original: We note that the use oflocal Markov chains is well motivated and preferred in theory and practice for the, in principle, lowcomputation time for implementing a single step.
Revised: We note that the use of local Markov chains is well motivated and preferred in theory and practice for the, in principle, low computation time for implementing a single step.

Original: For this reason, in most (discrete-state) cases thelocality parameter ∆ > 0 is tuned so that the ∆ -neighborhood of each point is at most of polynomialsize; see e.g. [Jer92, BWZ20, GZ19, CMZ22].
Revised: For this reason, in most (discrete-state) cases the locality parameter ∆ > 0 is tuned so that the ∆ -neighborhood of each point is at most of polynomial size; see e.g. [Jer92, BWZ20, GZ19, CMZ22].

Original: We now state the core result of this section, followed by various corollaries.
Revised: We now state the core result of this section, followed by various corollaries.

Original: (See the supplementarymaterial for a slightly more general result and for the proof).
Revised: (See the supplementary material for a slightly more general result and for the proof).

Original: The proof is based on an free energybarrier argument used in [BGJ20] which relies on rotational invariance of the Gaussian distribution.
Revised: The proof is based on an free energy barrier argument used in [BGJ20] which relies on rotational invariance of the Gaussian distribution.

Original: Theorem 5.3 (FP-Hard Implies Free Energy Barrier) .
Revised: Theorem 5.3 (FP-Hard Implies Free Energy Barrier) .

Original: Let µ be the uniform measure on S , where
Revised: Let µ be the uniform measure on S , where

Original: S ⊆ S N − 1 is a ﬁnite, transitive-symmetric set.
Revised: S ⊆ S N − 1 is a ﬁnite, transitive-symmetric set.

Original: The following holds for any ε ∈ (0 , 1 / 2) , D ≥ 2 ,λ ≥ 0 , and β ≥ 0 .
Revised: The following holds for any ε ∈ (0 , 1 / 2) , D ≥ 2 , λ ≥ 0 , and β ≥ 0 .

Original: Fix a ground-truth signal u ∈ S and let Y
Revised: Fix a ground-truth signal u ∈ S and let Y

Original: =
Revised: =

Original: λu + Z with Z ∼ N (0 , I N ) .
Revised: λu + Z with Z ∼ N (0 , I N ) .

Original: Deﬁneδ =
Revised: Deﬁne δ =

Original: δ ( D ) as in (3) .
Revised: δ ( D ) as in (3) .

Original: Let A = { v ∈ S : |(cid:104) u, v (cid:105)| ≤ δ } and B = { v ∈ S :
Revised: Let A = { v ∈ S : |(cid:104) u, v (cid:105)| ≤ δ } and B = { v ∈ S :

Original: (cid:104) u, v (cid:105) ∈
Revised: (cid:104) u, v (cid:105) ∈

Original: ( δ, (1 + ε ) δ ] } .
Revised: ( δ, (1 + ε ) δ ] } .

Original: With probability at least 1 − e − εD over Z , the Gibbs measure (8) associated to Y satisﬁes thatν β ( B ) ν β ( A ) ≤ 2 (cid:16) 2 · FP( D + log 2 , ˜ λ ) (cid:17) 1 − 2 ε e − εD where ˜ λ := (cid:113) βλ · 2+ ε 1 − 2 ε .
Revised: With probability at least 1 − e − εD over Z , the Gibbs measure (8) associated to Y satisﬁes that [Equation] where ˜ λ := (cid:113) βλ · 2+ ε 1 − 2 ε .

Original: As mentioned above, one particularly natural choice of β is the Bayesian temperature λ (whichcorresponds to sampling from the posterior distribution).
Revised: As mentioned above, one particularly natural choice of β is the Bayesian temperature λ (which corresponds to sampling from the posterior distribution).

Original: By combining Proposition 5.2 and Theorem 5.3 we conclude that FP-hard implies that the hitting time grows exponential with D, which wesimply refer to as “MCMC-hard”.
Revised: By combining Proposition 5.2 and Theorem 5.3 we conclude that FP-hard implies that the hitting time grows exponential with D, which we simply refer to as “MCMC-hard”.

Original: Combinining now this result with the FP and LD equivalence partdescribed in Corollary 4.5, we conclude also that LD-hard implies MCMC-hard.
Revised: Combinining now this result with the FP and LD equivalence part described in Corollary 4.5, we conclude also that LD-hard implies MCMC-hard.

Original: Corollary 5.4 (FP-Hard (and LD-hard) Implies Hitting Time Lower Bound) .
Revised: Corollary 5.4 (FP-Hard (and LD-hard) Implies Hitting Time Lower Bound) .

Original: In the setting of
Revised: In the setting of

Original: Theorem 5.3, suppose X 0 , X 1 , X 2 , . .
Revised: Theorem 5.3, suppose X 0 , X 1 , X 2 , . .

Original: . is a ∆ -local Markov chain with state space S and stationarydistribution ν β , for some ∆ ≤ εδ .
Revised: . is a ∆ -local Markov chain with state space S and stationary distribution ν β , for some ∆ ≤ εδ .

Original: Deﬁne the hitting time τ := inf { t ∈ N : (cid:104) u, X t (cid:105) > δ } .
Revised: Deﬁne the hitting time τ := inf { t ∈ N : (cid:104) u, X t (cid:105) > δ } .

Original: Withprobability at least 1 − e − εD over Z , there exists a state v ∈
Revised: With probability at least 1 − e − εD over Z , there exists a state v ∈

Original: A such that for the initialization X 

Revised: A such that for the initialization X

Original: = v ,with probability at least 1 − e − εD/ 2 over the Markov chain,
Revised: = v , with probability at least 1 − e − εD/ 2 over the Markov chain,

Original: [Equation]
Revised: [Equation]

Original: If furthermore D = D n is a sequence with D = ω (1) such that D
Revised: If furthermore D = D n is a sequence with D = ω (1) such that D

Original: + log 2 is an even integer
Revised: + log 2 is an even integer

Original: LD( D + log 2 , (1 + ε )
Revised: LD( D + log 2 , (1 + ε )

Original: ˜ λ )
Revised: ˜ λ )

Original: ≤ B for a constant B > 0 , then there is a constant C = C ( B, (cid:15) ) > 0 forwhich τ ≥ C ( B, ε ) e εD/ 2 .
Revised: ≤ B for a constant B > 0 , then there is a constant C = C ( B, (cid:15) ) > 0 for which τ ≥ C ( B, ε ) e εD/ 2 .

Original: We note that Corollary 5.4 applies for all ∆ ≤ εδ ( D ) .
Revised: We note that Corollary 5.4 applies for all ∆ ≤ εδ ( D ) .

Original: For various models of interest, we notethat the range ∆ ≤ εδ ( D ) for the locality parameter ∆ contains the “reasonable” range of valueswhere the ∆ -neighborhoods are of polynomial size.
Revised: For various models of interest, we note that the range ∆ ≤ εδ ( D ) for the locality parameter ∆ contains the “reasonable” range of values where the ∆ -neighborhoods are of polynomial size.

Original: For example, let us focus on the well-studiedtensor PCA setting with a Rademacher signal and even tensor power, that is S
Revised: For example, let us focus on the well-studied tensor PCA setting with a Rademacher signal and even tensor power, that is S

Original: = { u = x ⊗ 2 p : x ∈{− n − p , n − p } n } .
Revised: = { u = x ⊗ 2 p : x ∈ {− n − p , n − p } n } .

Original: Then for any D > 0 and any ε > 0 , we have (cid:15)δ ( D ) =
Revised: Then for any D > 0 and any ε > 0 , we have (cid:15)δ ( D ) =

Original: Ω( n − p ) and therefore theεδ ( D ) -neighborhood of any point u
Revised: Ω( n − p ) and therefore the εδ ( D ) -neighborhood of any point u

Original: = x ⊗ 2 p ∈ S, contains all n ω (1) points u
Revised: = x ⊗ 2 p ∈ S, contains all n ω (1) points u

Original: = y ⊗ 2 p ∈ S where yhas any Hamming distance o ( √ n ) from x.
Revised: = y ⊗ 2 p ∈ S where y has any Hamming distance o ( √ n ) from x.

Original: Remark 5.5.
Revised: Remark 5.5.

Original: We note that the original work of [BGJ20], on which the proof of Theorem 5.3 isbased, showed failure of MCMC methods in a strictly larger (by a power of n ) range of λ than thelow-degree-hard regime for the tensor PCA problem.
Revised: We note that the original work of [BGJ20], on which the proof of Theorem 5.3 is based, showed failure of MCMC methods in a strictly larger (by a power of n ) range of λ than the low-degree-hard regime for the tensor PCA problem.

Original: In contrast, our MCMC lower bound usesthe same argument but matches the low-degree threshold (at the Bayesian temperature).
Revised: In contrast, our MCMC lower bound uses the same argument but matches the low-degree threshold (at the Bayesian temperature).

Original: This isbecause [BGJ20] only considers temperatures that are well above the Bayesian one: in their notation,their result is for constant β whereas the Bayesian β grows with n .
Revised: This is because [BGJ20] only considers temperatures that are well above the Bayesian one: in their notation, their result is for constant β whereas the Bayesian β grows with n .

Original:  Sparse Linear Regression
Revised:  Sparse Linear Regression

Original: In this section we prove sharp low-degree lower bounds for the hypothesis testing version of aclassical inference problem: sparse linear regression.
Revised: In this section we prove sharp low-degree lower bounds for the hypothesis testing version of a classical inference problem: sparse linear regression.

Original: This is signiﬁcant for a number of reasons:(i) we uncover a new computational phase transition in sparse regression, (ii) our proof involvesupper-bounding LD in terms of FP, illustrating that an FP-to-LD connection can be achieved outsidethe Gaussian additive model, and (iii) the proof requires a ﬁrst-of-its-kind conditional low-degreecalculation which seems difﬁcult to carry out directly without using FP, illustrating that FP can be apowerful tool for proving low-degree lower bounds.
Revised: This is signiﬁcant for a number of reasons: (i) we uncover a new computational phase transition in sparse regression, (ii) our proof involves upper-bounding LD in terms of FP, illustrating that an FP-to-LD connection can be achieved outside the Gaussian additive model, and (iii) the proof requires a ﬁrst-of-its-kind conditional low-degree calculation which seems difﬁcult to carry out directly without using FP, illustrating that FP can be a powerful tool for proving low-degree lower bounds.

Original: We consider the following detection task.
Revised: We consider the following detection task.

Original: Deﬁnition 6.1 (Sparse Linear Regression: Hypothesis Testing) .
Revised: Deﬁnition 6.1 (Sparse Linear Regression: Hypothesis Testing) .

Original: Given a sample size m ∈ N , featuresize n ∈ N , sparsity level k ∈ N with k ≤ n and noise level σ > 0 , we consider hypothesis testingbetween the following two distributions over ( X, Y ) ∈ R m × n × R m .
Revised: Given a sample size m ∈ N , feature size n ∈ N , sparsity level k ∈ N with k ≤ n and noise level σ > 0 , we consider hypothesis testing between the following two distributions over ( X, Y ) ∈ R m × n × R m .

Original:  Q generates a pair ( X, Y ) where both X and Y have i.i.d.
Revised: • Q generates a pair ( X, Y ) where both X and Y have i.i.d.

Original: N (0 , 1) entries.• P generates a pair ( X, Y ) as follows.
Revised: N (0 , 1) entries. • P generates a pair ( X, Y ) as follows.

Original: First X is drawn with i.i.d.
Revised: First X is drawn with i.i.d.

Original: N (0 , 1) entries.
Revised: N (0 , 1) entries.

Original: Then, fora planted signal u ∈ { 0 , 1 } n drawn uniformly from all binary vectors of sparsity exactlyk , and independent noise W ∼ N (0 , σ 2 I m ) , we set Y =
Revised: Then, for a planted signal u ∈ { 0 , 1 } n drawn uniformly from all binary vectors of sparsity exactly k , and independent noise W ∼ N (0 , σ 2 I m ) , we set Y =

Original: ( k + σ 2 ) − 1 / 2 ( Xu + W ) .
Revised: ( k + σ 2 ) − 1 / 2 ( Xu + W ) .

Original: Thisnormalization ensures Y ∼ N (0 , I m ) .
Revised: This normalization ensures Y ∼ N (0 , I m ) .

Original: In the following result, we provide rigorous “low-degree evidence” (and a matching upper bound) forthe precise threshold separating the “easy” and “hard” regimes.
Revised: In the following result, we provide rigorous “low-degree evidence” (and a matching upper bound) for the precise threshold separating the “easy” and “hard” regimes.

Original: Theorem 6.2.
Revised: Theorem 6.2.

Original: For θ ∈ (0 , 1) , deﬁne
Revised: For θ ∈ (0 , 1) , deﬁne

Original: [Equation]
Revised: [Equation]

Original: Consider sparse linear regression (Deﬁnition 6.1) in the following scaling regime: for ﬁxed constantsθ ∈ (0 , 1) and R > 0 , take n → ∞ with k = n θ + o (1) , σ 2{+
+} = o ( k ) , and m = (1 + o (1))
Revised: Consider sparse linear regression (Deﬁnition 6.1) in the following scaling regime: for ﬁxed constants θ ∈ (0 , 1) and R > 0 , take n → ∞ with k = n θ + o (1) , σ {+
+} = o ( k ) , and m = (1 + o (1))

Original: Rk log( n/k ) .
Revised: Rk log( n/k ) .

Original: (a) (Hard regime)
Revised: (a) (Hard regime)

Original: If R < R LD ( θ ) then no degree- o ( k ) polynomial f : R m ( n +1)
Revised: If R < R LD ( θ ) then no degree- o ( k ) polynomial f : R m ( n +1)

Original: → R achievesthe following notion of “strong separation”:
Revised: → R achieves the following notion of “strong separation”:

Original: [Equation]
Revised: [Equation]

Original: (b) (Easy regime)
Revised: (b) (Easy regime)

Original: If R > R LD ( θ ) then there is a polynomial-time algorithm for strong detection between P and Q (see Deﬁnition 2.2).
Revised: If R > R LD ( θ ) then there is a polynomial-time algorithm for strong detection between P and Q (see Deﬁnition 2.2).

Original: In line with existing low-degree lower bounds in the literature, the notion of strong separation ruled outin part (a) is a natural notion of success for polynomial-based tests: by Chebyshev’s inequality, (10)implies that strong detection is possible by thresholding f .
Revised: In line with existing low-degree lower bounds in the literature, the notion of strong separation ruled out in part (a) is a natural notion of success for polynomial-based tests: by Chebyshev’s inequality, (10) implies that strong detection is possible by thresholding f .

Original: Part (a) is interpreted as evidence thatwhen R < R LD ( θ ) , strong detection requires runtime exp(˜Ω( k )) .
Revised: Part (a) is interpreted as evidence that when R < R LD ( θ ) , strong detection requires runtime exp(˜Ω( k )) .

Original: This is tight, matching the runtimeof a brute-force search algorithm, which succeeds for any constants θ ∈ (0 , 1) and R > 0 (i.e., theproblem is always information-theoretically possible in our regime)
Revised: This is tight, matching the runtime of a brute-force search algorithm, which succeeds for any constants θ ∈ (0 , 1) and R > 0 (i.e., the problem is always information-theoretically possible in our regime)

Original: [RXZ21].
Revised: [RXZ21].

Original: The algorithm that gives the matching upper bound in part (b) is fairly simple but somewhat subtle:letting X j denote the j th column of X , the idea is to count the number of indices j ∈ [ n ] for which(cid:104) X j , Y (cid:105) / (cid:107) Y (cid:107) 2 exceeds a particular (carefully chosen) threshold, and then threshold this count.
Revised: The algorithm that gives the matching upper bound in part (b) is fairly simple but somewhat subtle: letting X j denote the j th column of X , the idea is to count the number of indices j ∈ [ n ] for which (cid:104) X j , Y (cid:105) / (cid:107) Y (cid:107) 2 exceeds a particular (carefully chosen) threshold, and then threshold this count.

Original: We note that prior work has also considered the related task of recovering u , given ( X, Y ) ∼ P(e.g. [Wai09, DT10, GZ22, RXZ19, RXZ21]).
Revised: We note that prior work has also considered the related task of recovering u , given ( X, Y ) ∼ P (e.g. [Wai09, DT10, GZ22, RXZ19, RXZ21]).

Original: Our result gives evidence that the existing algorithmsfor recovery cannot be improved in the regime θ ↓ 0 .
Revised: Our result gives evidence that the existing algorithms for recovery cannot be improved in the regime θ ↓ 0 .

Original: See the supplement for further discussion.
Revised: See the supplement for further discussion.

Original: Proof techniques and beyond sparse regression Proving the sharp low-degree lower bound inthe regime θ < 1 / 4 requires a conditional low-degree calculation: instead of bounding LD fortesting P versus Q , we bound LD for testing the conditional distribution P | A versus Q , for somehigh-probability event A .
Revised: Proof techniques and beyond sparse regression Proving the sharp low-degree lower bound in the regime θ < 1 / 4 requires a conditional low-degree calculation: instead of bounding LD for testing P versus Q , we bound LD for testing the conditional distribution P | A versus Q , for some high-probability event A .

Original: This is necessary because the standard LD blows up at a sub-optimalthreshold due to a rare “bad” event under P .
Revised: This is necessary because the standard LD blows up at a sub-optimal threshold due to a rare “bad” event under P .

Original: Conditioning arguments of this type are common forinformation-theoretic lower bounds (see e.g. [BMNN16, BMV + 17, PWB16, PWBM18]), but thisis (to our knowledge) the ﬁrst instance where conditioning has been needed for a low-degree lowerbound.
Revised: Conditioning arguments of this type are common for information-theoretic lower bounds (see e.g. [BMNN16, BMV + 17, PWB16, PWBM18]), but this is (to our knowledge) the ﬁrst instance where conditioning has been needed for a low-degree lower bound.

Original: We note that [Arp21] gave low-degree lower bounds for sparse regression by analyzing thestandard (non-conditioned) LD, and our result improves the threshold when θ < 1 / 4 via conditioning.
Revised: We note that [Arp21] gave low-degree lower bounds for sparse regression by analyzing the standard (non-conditioned) LD, and our result improves the threshold when θ < 1 / 4 via conditioning.

Original: The standard approach to bounding LD involves direct moment calculations (see e.g. Section 2.4of [Hop18]), but it seems difﬁcult to carry out our conditional low-degree calculation by this approach.
Revised: The standard approach to bounding LD involves direct moment calculations (see e.g. Section 2. of [Hop18]), but it seems difﬁcult to carry out our conditional low-degree calculation by this approach.

Original: Instead, we prove that the conditional LD can be upper-bounded by the corresponding conditional FP,which is more tractable to bound directly.
Revised: Instead, we prove that the conditional LD can be upper-bounded by the corresponding conditional FP, which is more tractable to bound directly.

Original: This illustrates that FP can be a powerful tool for provinglow-degree lower bounds that may otherwise be out of reach.
Revised: This illustrates that FP can be a powerful tool for proving low-degree lower bounds that may otherwise be out of reach.

Original: As we discuss in the supplement,the FP-to-LD connection we establish here holds for a broad class of problems with planted sparsesignals, including planted clique, planted dense subgraph, and any sparse generalized linear model.
Revised: As we discuss in the supplement, the FP-to-LD connection we establish here holds for a broad class of problems with planted sparse signals, including planted clique, planted dense subgraph, and any sparse generalized linear model.

Original:  Conclusion
Revised:  Conclusion

Original: We have introduced the Franz-Parisi criterion and have shown that it implies lower bounds againstlocal MCMC algorithms and is equivalent to low-degree hardness in Gaussian additive models.
Revised: We have introduced the Franz-Parisi criterion and have shown that it implies lower bounds against local MCMC algorithms and is equivalent to low-degree hardness in Gaussian additive models.

Original: We have also used the FP criterion as a tool to prove new low-degree lower bounds for sparseregression.
Revised: We have also used the FP criterion as a tool to prove new low-degree lower bounds for sparse regression.

Original: This is a ﬁrst step in relating notions of computational intractability based on free energypotentials to lower bounds within restricted models of computation, such as the sum-of-squareshierarchy.
Revised: This is a ﬁrst step in relating notions of computational intractability based on free energy potentials to lower bounds within restricted models of computation, such as the sum-of-squares hierarchy.

Original: A clear question in the wake of our work is whether similar connections can be drawn fora broader class of planted models.
Revised: A clear question in the wake of our work is whether similar connections can be drawn for a broader class of planted models.

Original: In the supplementary material, we demonstrate that this cannot bedone in a straightforward way, by building counterexamples (namely, certain simple mixtures over
Revised: In the supplementary material, we demonstrate that this cannot be done in a straightforward way, by building counterexamples (namely, certain simple mixtures over

Original: Boolean product measures) where the Franz-Parisi criterion is not consistent with computationalintractability.
Revised: Boolean product measures) where the Franz-Parisi criterion is not consistent with computational intractability.

Original: Nonetheless, given the impressive track-record of accuracy of free-energy basedcomputational threshold predictions, we believe this to be a shortcoming of our speciﬁc criterion,and the development of one for a more general class of problems is a fascinating direction for futureresearch.
Revised: Nonetheless, given the impressive track-record of accuracy of free-energy based computational threshold predictions, we believe this to be a shortcoming of our speciﬁc criterion, and the development of one for a more general class of problems is a fascinating direction for future research. Acknowledgments and Disclosure of Funding

Original: Checklist
Revised: AEA: Part of this work was done while this author was supported by the Richard M. Karp Fellowship at the Simons Institute for the Theory of Computing (Program on Probability, Geometry and Computation in High Dimensions). This author is grateful to Florent Krzakala for introducing him to the work of Franz and Parisi. SBH: Parts of this work were done while this author was supported by a Microsoft Research PhD Fellowship, by a Miller Postdoctoral Fellowship, and by the Microsoft Fellowship at the Simons

Revised: Institute for the Theory of Computing.

Revised: ASW:

Revised: Part of this work was done at Georgia Tech, supported by NSF grants CCF-2007443 and

Revised: CCF-2106444.

Revised: Part of this work was done while visiting the Simons Institute for the Theory of

Revised: Computing.

Revised: Part of this work was done while with the Courant Institute at NYU, partially supported by NSF grant DMS-1712730 and by the Simons Collaboration on Algorithms and Geometry.

Revised: IZ: Supported by the Simons-NSF grant DMS-2031883 on the Theoretical Foundations of Deep

Revised: Learning and the Vannevar Bush Faculty Fellowship ONR-N00014-20-1-2826.

Revised: Part of this work was done while visiting the Simons Institute for the Theory of Computing.

Original: For all authors...
Revised: Part of this work was done while with the Center for Data Science at NYU, supported by a Moore-Sloan CDS postdoctoral fellowship. The authors thank Cris Moore and the Santa Fe Institute for hosting the 2018 “Santa Fe Workshop on Limits to Inference in Networks and Noisy Data,” where the initial ideas in this paper were formulated. The authors thank Aukosh Jagannath and anonymous reviewers for helpful comments on an earlier version of this work.

Original:  (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s contributions and scope?
Revised: For all authors... (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s contributions and scope?

Original: [Yes] (b) Did you describe the limitations of your work?
Revised: [Yes] (b) Did you describe the limitations of your work?

Original: [Yes] (c) Did you discuss any potential negative societal impacts of your work?
Revised: [Yes] (c) Did you discuss any potential negative societal impacts of your work?

Original: [N/A]
Revised: [N/A]

Original: Our work is purely theoretical and has no societal impacts.
Revised: Our work is purely theoretical and has no societal impacts.

Original: (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?
Revised: (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?

Original: [Yes] 
Revised: [Yes] 2.

Original: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?
Revised: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes] (b) Did you include complete proofs of all theoretical results?
Revised: [Yes] (b) Did you include complete proofs of all theoretical results?

Original: [Yes]
Revised: [Yes]

Original: In the supplementary material.
Revised: In the supplementary material.

Original: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)?
Revised: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?

Original: [N/A] (b) Did you specify all the training details (e g., data splits, hyperparameters, how theywere chosen)?
Revised: [N/A] (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?

Original: [N/A] (c) Did you report error bars (e g., with respect to the random seed after running experi- ments multiple times)?
Revised: [N/A] (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?

Original: [N/A] (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?
Revised: [N/A] (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?

Original: [N/A] 
Revised: [N/A] 4.

Original: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?
Revised: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?

Original: [N/A] (b) Did you mention the license of the assets?
Revised: [N/A] (b) Did you mention the license of the assets?

Original: [N/A] (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: [N/A] (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [N/A] (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?
Revised: [N/A] (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?

Original: [N/A] (e) Did you discuss whether the data you are using/curating contains personally identiﬁable information or offensive content?
Revised: [N/A] (e) Did you discuss whether the data you are using/curating contains personally identiﬁable information or offensive content?

Original: [N/A] 
Revised: [N/A] 5.

Original: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?
Revised: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?

Original: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?
Revised: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?

Original: [N/A] (c) Did you include the estimated hourly wage paid to participantsand the total amount spent on participant compensation? [N/A]
Revised: [N/A] (c) Did you include the estimated hourly wage paid to participants