{"id":"28aaea1a-a6a5-401a-9af7-b652124d2087","arxiv_id":"1908.07460","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The minimax rate for estimating μ^T Σ^{-1} μ under sparsity of Σ^{-1} μ is (s log p)/n + 1/√n, attained by a debiased ℓ1-regularized plug-in estimator.","lead":"This paper finds the best possible accuracy for estimating the quantity μ^T Σ^{-1} μ, which measures portfolio performance and class separability, when there are many variables. It provides an estimator that reaches that accuracy and shows that simpler plug-in estimators do not.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The §3.4 approximate-sparsity rates as displayed fail their own q=0 check: on H₀(R,τ)=H(R,τ), Corollary 2(b) gives (1+τ)^{1/2}R(logp/n)^{1/2} with transition τ≍R√(logp/n), contradicting Corollary 1(b)'s (1+τ)slogp/n with transition τ≍slogp/n.","rationale":"I read the exact-sparse argument in good faith and it holds up: the lower-bound constructions (mean-shift and covariance-shift two-point tests for the parametric term; chi-squared two-fuzzy-hypotheses for the sparse term) are valid under the stated scalings, the KKT/cone arguments in Lemma 3–4 and the complement-event bounds in the proof of Theorem 1 are coherent, and the phase-transition matching between (3.9) and (3.10) checks out in all τ regimes. The reader's flagged concern (sparsity and eigenvalue assumptions) is a genuine scope caveat, but Proposition 1 shows the sparsity assumption is necessary, and within H(s,τ) the central claim is supported. My concern is different and more specific: the approximate-sparsity generalization of Section 3.4, as displayed, fails its own q=0 recovery test. Since H₀(R,τ)=H(R,τ), Corollary 2(b) at q=0 must equal Corollary 1(b); instead it asserts a rate (1+τ)^{1/2}R(logp/n)^{1/2} and a transition at R√(logp/n), contradicting the exact-sparse minimax rate (1+τ)slogp/n and transition at slogp/n. The proof's own effective-sparsity substitution suggests the intended exponents are 1−q/2 rather than (1−q)/2, so the underlying technique is probably sound and the displayed formulas are mis-set; but as printed, one of Corollary 1(b) or Corollary 2(b) must be wrong, and the assertion that q=0 fully recovers the exact results is false. That warrants a conditional verdict: accept the exact-sparse core, require the Section 3.4 exponents to be corrected and re-verified before the generalization claims are taken as established.","tokens_in":56051,"tokens_out":40554,"duration_ms":362003,"concrete_test":"Set q=0 and R=s in Theorem 4, Theorem 5(b), and Corollary 2(b), and compare the displayed rates and phase-transition thresholds term by term with Corollary 1(a)-(b) and Theorem 2(b) on the identical class H₀(R,τ)=H(R,τ). A correct specialization must match the exact-sparse results exactly. If the (1−q)/2 exponents fail to reproduce (1+τ)s log p/n and τ≍s log p/n, then re-derive Theorem 4's final bound from the proof's effective-sparsity substitution s_eff = R̃(1+τ̃)^{−q/2}(logp/n)^{−q/2} together with Lemma 6 and the cone/basic-inequality steps, determine whether the correct rate is (1+τ̃)^{1−q/2}R̃(logp/n)^{1−q/2}, correct Corollary 2's phase-transition threshold accordingly, and re-verify that the corrected Theorem 5(b) lower bound matches the corrected upper bound in each of the three τ̃ regimes.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The exact-sparse core (Theorems 1–2, Corollary 1) is internally coherent: the two-point constructions for the τ∧(τ+√τ)/√n term, the chi-squared two-fuzzy-hypotheses calculations for the s log p/n term in Section 5.3, and the cone/KKT bounds of Lemmas 3–4 are mutually consistent and match the displayed rates in all three τ regimes. The load-bearing problem is the claimed generalization in Section 3.4. Theorem 4 and Corollary 2(b) display the approximate-sparsity rate τ̃∧[(1+τ̃)^{(1−q)/2}R̃(logp/n)^{(1−q)/2}] and the phase transition at τ̃ ≍ R̃(1+τ̃)^{−q/2}(logp/n)^{(1−q)/2}. Because H₀(R,τ) with the paper's own 0⁰=0 convention coincides with the exact-sparse class H(R,τ), setting q=0 in these formulas must reproduce Corollary 1 exactly. It does not: q=0 yields (1+τ)^{1/2}R(logp/n)^{1/2} with transition at τ≍R√(logp/n), whereas Corollary 1(b) and Theorem 2(b) give [τ∧(1+τ)slogp/n]+[τ∧(τ+√τ)/√n] with transition at τ≍slogp/n. The sentence 'Setting q = 0 in the two theorems, we fully recover (3.7)–(3.10)' is therefore false as stated. The proof of Theorem 5(b)(i) itself substitutes effective sparsity s_eff = R̃(logp/n)^{−q/2}τ̃^{−q/2} into the exact-sparse chi-squared construction, which yields the different rate τ̃^{1−q/2}R̃(logp/n)^{1−q/2} — consistent with exponents 1−q/2, not (1−q)/2. So either the displayed exponents are mis-set or Corollary 2 is incorrect; as the manuscript stands, Corollary 2(b) and Corollary 1(b) assert different minimax rates for the same parameter space, so the generalization's rates are not established as printed.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies minimax estimation of the quadratic functional θ = μ^T Σ^{-1} μ from n i.i.d. sub-Gaussian vectors in R^p, under the assumption that α = Σ^{-1} μ is exactly or approximately sparse and that the eigenvalues of Σ are bounded. It proposes a debiased ℓ1-regularized plug-in estimator (3.1)–(3.2), proves upper bounds (Theorem 1), matching lower bounds (Theorem 2), and establishes a phase transition at τ ≍ s log p/n (Corollary 1). It further shows that a family of naive plug-in estimators with c ≠ 2 is suboptimal (Proposition 2), studies an ℓ0 variant (Theorem 3), extends the framework to robust inputs of mean and covariance estimators (Section 3.3), to approximate sparsity classes (Section 3.4), and to the dense regime (Section 3.5), and reports simulation and S&P500 portfolio experiments (Section 4).","tokens_in":56581,"tokens_out":13845,"duration_ms":124306,"significance":"If the results stood as stated, the paper would make a substantial contribution to high-dimensional functional estimation with unknown covariance, extending Fan, Rigollet, and Wang (2015) and Collier, Comminges, and Tsybakov (2017) to the setting where Σ is unknown and α = Σ^{-1} μ is sparse. The strengths are the complete proofs in Section 5, the explicit two-point and χ² lower-bound constructions, the feasible debiased estimator that attains the exact-sparse rate, and the detailed treatment of the dense regime. However, the approximate-sparsity generalization in Section 3.4 is internally inconsistent with the exact-sparse results when q = 0, and the exponential factor in Theorem 2(b) prevents the claimed Corollary 1(b) lower bound as written. These defects are localized and likely reparable, but they affect two advertised central claims, so the paper needs a major revision before acceptance.","major_comments":[{"comment":"The claim that setting q = 0 in the approximate-sparsity results recovers the exact-sparse results is false with the displayed formulas. Since H_0(R, τ) = H(R, τ) under the paper's 0^0 = 0 convention, Corollary 2(b) with q = 0 must coincide with Corollary 1(b). Instead, q = 0 in Corollary 2(b) yields (1 + τ)^{1/2} R (log p/n)^{1/2} with a phase transition at τ ≍ R (log p/n)^{1/2}, whereas Corollary 1(b) yields [τ ∧ (1 + τ) s log p/n] + [τ ∧ (τ + √τ)/√n] with a transition at τ ≍ s log p/n. The proof in Section 5.7.1 shows which rate the lower-bound construction actually supports: in case (i) the proof substitutes the effective sparsity s_eff = R̃ (log p/n)^{-q/2} τ̃^{-q/2} into the exact-sparse χ² construction and obtains a rate of order τ̃^{1-q/2} R̃ (log p/n)^{1-q/2}, consistent with exponents 1 - q/2, not (1 - q)/2. Thus the displayed exponents in Theorem 4, Theorem 5, and Corollary 2 need to be corrected, and the sentence 'Setting q = 0 in the two theorems, we fully recover (3.7)–(3.10)' must be revised accordingly. The scaling conditions in Theorem 4 and Corollary 2 should also be rechecked after this correction.","section":"Theorem 2(b) and the derivation of Corollary 1(b)"},{"comment":"The lower bound in Theorem 2(b) contains the factor c_0 exp(-e^{2s^2 p^{c_6 c_0^{-1}}}) with c_0 ∈ [0,1] and c_6 > 0. For p ≥ 2 and s ≥ 1, this factor is exponentially small for every admissible c_0, and the argument after (3.10) that one can 'choose sufficiently small c_0' to make e^{2s^2 p^{c_6 c_0^{-1}} = 1 + o(1) is not valid; decreasing c_0 increases the positive exponent c_6 c_0^{-1}. As stated, the second displayed term in Theorem 2(b) therefore does not yield the claimed [τ ∧ (1 + τ) s log p/n] lower bound in Corollary 1(b). The authors need to restate the exponential factor (for example, with a negative exponent on p, if that is the intended expression) and the precise condition under which the factor is bounded below by a universal constant.","section":"Theorem 2(b) and the derivation of Corollary 1(b)"}],"minor_comments":[{"comment":"The phrase 'Shape ratio' in the discussion of Table 1 should be 'Sharpe ratio'.","section":"Section 4.2"},{"comment":"The reuse of the symbol c both as an arbitrary positive constant in the exponent and as the constant controlling the scaling condition involving c̃ is confusing; please use distinct symbols.","section":"Section 3.1, Theorem 1"},{"comment":"The condition (3.17) involves an unspecified sample size m; in the applications to the one-sample and two-sample problems, please clarify whether m is n, n1, or n2 and how it enters the subsequent displayed bounds.","section":"Section 3.3, Proposition 3"},{"comment":"The theoretical rate functions f_α, g_α, h_α and f_θ, g_θ, h_θ are plotted with calibrated constants, but the text does not explicitly state which theorem or corollary each dashed curve corresponds to; adding this mapping would improve reproducibility.","section":"Section 4.1, Figure 1"},{"comment":"In the proof of Theorem 2(b), the chi-squared calculations are stated to follow from Lemma A.1 of Fan, Rigollet, and Wang (2015); since this lemma is central to the lower bound, stating it explicitly in Section 5.9 would make the proof more self-contained.","section":"Section 5.3.1"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the exact-sparse core (Theorems 1–3, Corollary 1, Propositions 1–2) appears to be a solid and largely self-contained contribution, and the q = 0 inconsistency in Section 3.4 plus the problematic exponential factor in Theorem 2(b) seem reparable through careful restatement. I therefore do not recommend rejection; however, the approximate-sparsity rates and the exact-sparse lower bound as currently displayed are load-bearing claims that need to be corrected and reverified before the paper can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The exact-sparse core is the real contribution, and it holds up. Estimating θ = μᵀΣ⁻¹μ with unknown Σ and sparse α = Σ⁻¹μ is a nontrivial extension of Collier–Comminges–Tsybakov and Fan–Rigollet–Wang, and the paper gets the phase transition right: the minimax rate is [τ∧(τ+√τ)/√n] + [τ∧(1+τ)s log p/n], with the debiased ℓ₁ plug-in attaining it when τ ≳ s log p/n and the trivial estimator 0 being optimal below that. The two-point constructions, the chi-squared calculations in §5.3, and the cone/KKT bounds in Lemmas 3–4 are mutually consistent. I checked the main regimes and the matching is genuine, not cosmetic. The citation pattern is fine, and the admitted non-optimal terms in Theorem 1 and the opacity in Theorem 2(b) are minor blemishes, not red flags.\n\nThe problem is §3.4. The claim that setting q = 0 recovers the exact-sparse theorems is false as printed. For q = 0, H₀(R,τ) is exactly H(R,τ), but Corollary 2(b) gives (1+τ)^{1/2}R(log p/n)^{1/2} with transition at τ ≍ R√(log p/n), whereas Corollary 1(b) gives (1+τ)s log p/n with transition at τ ≍ s log p/n. The proof sketch for Theorem 5(b) actually substitutes effective sparsity s_eff = R̃(log p/n)^{-q/2}τ̃^{-q/2} into the exact-sparse chi-squared construction, which produces exponent 1−q/2 on τ and log p/n, not (1−q)/2. So either the displayed exponents are mis-set or Corollary 2 is wrong; both cannot be correct. This is not a constants quibble — the two formulas scale differently in log p and n. The generalization to approximate sparsity is therefore not established as written, and the paper's own q=0 consistency check fails.\n\nWho should read it: anyone working on high-dimensional functional estimation, portfolio optimization, or LDA will want the exact-sparse result, and the debiasing idea is clean and potentially reusable. But the approximate-sparsity section as it stands would mislead a careful reader.\n\nRecommendation: send it to a serious referee, but with the expectation of major revision. The exact-sparse half deserves publication; the approximate-sparsity half needs to be fixed — or honestly delimited — before the paper can be accepted.","headline":"The exact-sparse minimax results for μᵀΣ⁻¹μ are solid and worth serious refereeing, but the approximate-sparsity generalization in §3.4 is internally inconsistent and needs a major fix before publication.","tokens_in":57098,"tokens_out":7864,"would_cite":true,"duration_ms":78630,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H12","62C20","62F12"],"pacs":[],"model":"deepseek-v4-flash","headline":"Estimating μᵀΣ⁻¹μ has a sharp minimax rate, and the paper constructs an estimator that achieves it.","keywords":["functional estimation","high-dimensional statistics","quadratic functional","mean and covariance","minimax optimality","phase transition","sparsity","sub-gaussian distribution"],"falsifier":"Run the debiased estimator and the best $c\\ne 2$ plug-in on Gaussian data with $s$ equal nonzero entries in $\\alpha$, $\\tau$ constant, and $\\tau \\gg s\\log p/n$; if the plug-in's worst-case error falls below order $\\sqrt{\\tau(1+\\tau)}\\, s\\log p/n$, or the debiased error exceeds order $(\\tau+\\sqrt{\\tau})/\\sqrt{n} + (1+\\tau)s\\log p/n$, the rate statements are wrong.","tokens_in":55877,"feed_emoji":"📈","tokens_out":12361,"duration_ms":107151,"temperature":0.7,"pith_summary":"The paper asks how well one can estimate the single number $\\theta = \\mu^T \\Sigma^{-1} \\mu$ — the squared Sharpe ratio or the linear-discriminant signal — from $n$ high-dimensional samples when the vector $\\alpha = \\Sigma^{-1} \\mu$ is sparse. It proves that without sparsity no consistent estimator exists when $p \\ge n^2$, and that under sparsity the minimax absolute error is of order $[\\tau \\wedge (\\tau+\\sqrt{\\tau})/\\sqrt{n}] + [\\tau \\wedge (1+\\tau) s \\log p / n]$, up to constants. A bias-corrected plug-in estimator defined through an $\\ell_1$-regularized estimate of $\\alpha$ attains this rate whenever the signal $\\tau$ is above $s \\log p/n$; below that threshold the trivial estimator $0$ is already minimax optimal. Naive plug-in estimators with the same regularization fail, because the $\\ell_1$ bias contaminates the quadratic term. The result matters because $\\theta$ governs portfolio performance and classification difficulty, and knowing the exact rate tells practitioners when sparse estimation helps and when it cannot.","feed_headline":"Squared Sharpe ratio gets a sharp minimax rate","feed_subtitle":"A debiased ℓ1 plug-in attains the optimal error; naive plug-in estimators provably fall short.","key_machinery":"The central object is the bias-corrected plug-in estimator $\\tilde{\\theta} = 2\\hat{\\mu}^T \\tilde{\\alpha} - \\tilde{\\alpha}^T \\hat{\\Sigma} \\tilde{\\alpha}$, built from the $\\ell_1$-regularized estimator $\\tilde{\\alpha}$ defined by minimizing $\\frac12 \\beta^T\\hat{\\Sigma}\\beta - \\beta^T\\hat{\\mu} + \\lambda\\|\\beta\\|_1$ over a bounded ball. The estimator works because the KKT conditions for $\\tilde{\\alpha}$ express its main bias as $-\\lambda \\Sigma^{-1} \\hat{g}$, and replacing $\\Sigma^{-1}\\hat{\\mu}$ by $\\tilde{\\alpha}$ cancels that bias in the plug-in product. The proofs show that on a high-probability event the error $\\tilde{\\alpha}-\\alpha$ lies in a sparse cone where coordinates off the support are controlled by three times the on-support coordinates, which yields the $\\ell_2$ bound. Lower bounds come from two-point and fuzzy-hypothesis tests with $\\chi^2$-divergence computations for the matching minimax statements.","core_discovery":"The central discovery is that the estimation error for $\\theta$ undergoes a sharp phase transition between a parametric rate and a high-dimensional sparse rate. On the class $H(s,\\tau)$ where $\\alpha=\\Sigma^{-1}\\mu$ has at most $s$ nonzero entries and $\\theta\\le\\tau$, with eigenvalues of $\\Sigma$ between fixed constants, the minimax rate is $[\\tau \\wedge (\\tau+\\sqrt{\\tau})/\\sqrt{n}] + [\\tau \\wedge (1+\\tau)s \\log p / n]$. The bias-corrected estimator $\\tilde{\\theta} = 2\\hat{\\mu}^T \\tilde{\\alpha} - \\tilde{\\alpha}^T \\hat{\\Sigma} \\tilde{\\alpha}$, computed from an $\\ell_1$-regularized M-estimator $\\tilde{\\alpha}$ of $\\alpha$, achieves this rate and is minimax optimal when $\\tau \\gtrsim s\\log p/n$; when $\\tau \\lesssim s\\log p/n$ the trivial estimator $0$ is optimal. A family of plug-in estimators $\\tilde{\\theta}_c = c\\hat{\\mu}^T\\tilde{\\alpha} + (1-c)\\tilde{\\alpha}^T\\hat{\\Sigma}\\tilde{\\alpha}$ is suboptimal for every constant $c\\ne 2$, with error at least of order $\\sqrt{\\tau(1+\\tau)}\\, s\\log p / n$. In the dense regime $s\\gtrsim\\sqrt{p}$, a different de-biasing — an unbiased inverse-Wishart correction for Gaussian data and an iterated data-splitting bias correction for sub-Gaussian data — gives rate $\\sqrt{p}/n + (\\tau+\\sqrt{\\tau})/\\sqrt{n}$, completing an elbow at $s\\approx\\sqrt{p}$.","pith_inferences":["The paper's phase-transition analysis implies that reported Sharpe ratios or LDA signal estimates in high dimension should come with a regime flag: below $\\tau \\approx s\\log p/n$, the estimate is essentially indistinguishable from zero, and confidence statements need to reflect that.","Because the suboptimality of $c\\ne 2$ plug-ins is driven by $\\lambda\\|\\tilde{\\alpha}\\|_1$, any feasible estimator of $\\theta$ built from a sparse $\\alpha$ must either remove this bias or use an $\\ell_0$-type construction; a testable extension is to check whether Dantzig-selector plug-ins obey the same gap outside the paper's theory.","The open scaling $p\\lesssim s^2$ with $n\\lesssim p\\lesssim n^2$ suggests there should be an interpolation between the sparse and dense rates; a simulation study sweeping $(n,p,s,\\tau)$ across that boundary could reveal the missing minimax formula.","Since $\\theta$ controls the Bayes error in linear discriminant analysis, the minimax gap here quantifies precisely when sparse LDA can be expected to beat random guessing in high dimensions."],"forward_implications":["When the signal is strong enough ($\\tau \\gtrsim s\\log p/n$), the debiased estimator achieves the minimax rate, so no other estimator can do uniformly better over the sparse class.","When $\\tau \\lesssim s\\log p/n$, estimating by zero is optimal; this is a genuine regime where the signal is too weak for any data-based estimator to beat the trivial guess.","Naive plug-in estimators built from the same $\\ell_1$ estimate are provably suboptimal for every constant $c\\ne 2$, so the de-biasing step is rate-determining, not cosmetic.","In the dense regime $s\\gtrsim\\sqrt{p}$, the minimax rate becomes $(1+\\tau)\\sqrt{p}/n + (\\tau+\\sqrt{\\tau})/\\sqrt{n}$, giving an elbow at $s\\approx\\sqrt{p}$.","The same de-biased scheme with robust estimators of mean and covariance extends the rates (with an extra factor $s$ for estimating $\\alpha$) to heavy-tailed distributions with bounded fourth moments."],"supporting_citations":[{"why":"Supplies the phase-transition pattern and the chi-square fuzzy-hypothesis lower-bound machinery for functionals of sparse covariance matrices.","marker":"Fan, Rigollet, and Wang (2015)"},{"why":"Gives the minimax rates for quadratic functionals on sparsity classes in the Gaussian sequence model that this paper extends to unknown covariance.","marker":"Collier, Comminges, and Tsybakov (2017)"},{"why":"Introduces the de-biasing idea for high-dimensional linear models that motivates the estimator (3.2).","marker":"Zhang and Zhang (2014)"},{"why":"Provides the debiased lasso inference framework used to construct the corrected plug-in estimator.","marker":"Javanmard and Montanari (2014a,b)"},{"why":"Supplies the debiased M-estimator perspective underlying the bias correction in (3.2).","marker":"Van de Geer, Bühlmann, Ritov and Dezeure (2014)"},{"why":"Provides the unified M-estimator error analysis used to control the regularized estimator $\\tilde{\\alpha}$.","marker":"Negahban, Ravikumar, Wainwright, and Yu (2012)"},{"why":"Supplies the two-point and Fano/fuzzy-hypothesis testing inequalities used in the minimax lower bounds.","marker":"Tsybakov (2009)"},{"why":"Gives the high-dimensional Markowitz plug-in behavior and the unbiased estimator used as a benchmark in the dense regime.","marker":"Karoui (2010)"}],"fun_headline_variants":["Phase transition in optimal estimation of squared Sharpe ratio","Debiased plug-in hits minimax rate for sparse functional","Naive plug-in fails; debiased wins for high-dimensional functional","Sharp minimax rate for functional of mean and covariance"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything hinges on $\\Sigma^{-1}\\mu$ being sparse (or nearly so) while the eigenvalues of $\\Sigma$ stay between two fixed constants; without that, Proposition 1 shows the functional cannot be consistently estimated when $p \\ge n^2$.","fun_headline_variants_meta":{"raw":{"variants":["Phase transition in optimal estimation of squared Sharpe ratio","Debiased plug-in hits minimax rate for sparse functional","Naive plug-in fails; debiased wins for high-dimensional functional","Sharp minimax rate for functional of mean and covariance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001007,"raw_usage":{"total_tokens":4315,"prompt_tokens":1063,"completion_tokens":3252,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":679,"completion_tokens_details":{"reasoning_tokens":3184}},"tokens_in":679,"tokens_out":3252,"duration_ms":24483,"temperature":1.0,"reasoning_tokens":3184,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:17:44.766708+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the debiased estimator and the best $c\\ne 2$ plug-in on Gaussian data with $s$ equal nonzero entries in $\\alpha$, $\\tau$ constant, and $\\tau \\gg s\\log p/n$; if the plug-in's worst-case error falls below order $\\sqrt{\\tau(1+\\tau)}\\, s\\log p/n$, or the debiased error exceeds order $(\\tau+\\sqrt{\\tau})/\\sqrt{n} + (1+\\tau)s\\log p/n$, the rate statements are wrong.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the phase-transition pattern and the chi-square fuzzy-hypothesis lower-bound machinery for functionals of sparse covariance matrices."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the minimax rates for quadratic functionals on sparsity classes in the Gaussian sequence model that this paper extends to unknown covariance."},{"cited_title":"and Zhang, S","cited_arxiv_id":null,"evidence_quote":"Introduces the de-biasing idea for high-dimensional linear models that motivates the estimator (3.2)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the unified M-estimator error analysis used to control the regularized estimator $\\tilde{\\alpha}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the two-point and Fano/fuzzy-hypothesis testing inequalities used in the minimax lower bounds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the high-dimensional Markowitz plug-in behavior and the unbiased estimator used as a benchmark in the dense regime."}],"review_version":1}