{"id":"655bd98f-711b-475b-a5f1-8d5fd736845b","arxiv_id":"2607.16551","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Random basis expansions with non-continuous dense-support weight sampling have the universal approximation property with arbitrarily high probability.","lead":"This paper claims that random basis expansions can still approximate any continuous function with high probability when weights are sampled from a non-continuous distribution whose support is dense. It extends existing universal approximation results to complex-valued activations and sparse sampling schemes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1 is ill-posed: φ:R→C is applied to complex inputs ω^T x+b for x∈C^n, and the cited Voigtlaender theorem does not cover this setting.","rationale":"I reproduce the reader's central concern. The theorem's own statement sets the target on C^n but the architecture uses real affine maps into R with a real-domain activation. That is not a harmless typo: it makes the assertion 'φ_j approximates f' undefined, and the proof supplies no extension convention. The only cited result for the complex case, Theorem 2.3, is quoted as requiring φ:C→C; even taken at face value, it is not applicable to φ:R→C. Lemma 2.4, which is supposed to transfer closeness of weights to closeness of functions, explicitly operates on K⊂R^n and thus cannot bridge complex inputs. Therefore the main advertised extension—complex-valued activations in random basis expansions—rests on an unsupported identification.\n\nI also note the probability calculation has independent problems. In (11), the direction of the inequality is reversed relative to the required lower bound, and the displayed logarithmic bound in (12) does not follow from (10)–(11); this affects the claimed high-probability quantification even in the real case. The dense-support-to-positive-measure step is legitimate under the standard definition of support, but it could be stated explicitly.\n\nNone of this is an attack on the plausibility of a repairable theorem: restricting to x∈R^n with f:R^n→C, or switching to complex weights and φ:C→C, might restore the argument. But the paper as written claims f:C^n→C, and the proof's central application of Theorem 2.3 is invalid. This confirms the reader's REJECT; I recommend no change to that verdict.","tokens_in":5323,"tokens_out":7050,"duration_ms":73184,"concrete_test":"Consult Voigtlaender (2023) and verify the exact hypotheses of the theorem cited as [16, Theorem 2.3]: specifically whether it asserts universality for φ:C→C on compact subsets of R^n only, or also for φ:R→C with inputs in C^n. If the former, the step at equation (8) in the proof of Theorem 3.1 is unsupported, and the concern lands. To make it fully operational, instantiate Theorem 3.1 with n=1, f(z)=z on the unit disk, φ(t)=cos t, and evaluate φ_j(i/2); if no definition of φ_j at a complex input is supplied, the theorem is ill-posed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim fails as stated because the object in Theorem 3.1 is undefined. The theorem takes f:C^n→C and K⊂C^n, but defines basis functions φ_j(x)=φ(ω_j^T x+b_j) with (ω_j,b_j)∈R^{n+1} and φ:R→C. For any x with nonzero imaginary part, ω_j^T x+b_j is not a real number, so φ_j(x) is meaningless. The proof's bridge to the complex case is the sentence 'If f:C^n→C and φ:R→C, this follows from Theorem 2.3,' but the stated Theorem 2.3 (Voigtlaender) requires φ:C→C and, via Definition 2.1, only approximates functions on R^n; it says nothing about real-valued activations on C^n. Lemma 2.4, the only continuity argument, is also restricted to K⊂R^n. Thus equation (8), which supplies the approximating q-term expansion, has no valid hypothesis for the complex-input case. The advertised result is therefore not proved. Secondary issues—the direction of inequality (11)–(12) is inconsistent as written, and dense support gives v_i>0 only under the standard positive-neighborhood definition of support—reinforce the rejection but are not the main defect.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to prove a high-probability universal approximation theorem for random basis expansions whose weights are sampled from a distribution with dense support in R^{n+1}, allowing non-continuous sampling distributions and complex-valued activation functions. Theorem 3.1 states that for continuous f:C^n→C and a continuous non-polyharmonic φ:R→C, iid samples (ω_j,b_j) from such a distribution give, with high probability and with an explicit logarithmic width bound, an expansion Σ_j α_j φ(ω_j^T x + b_j) that uniformly approximates f on any compact K⊂C^n. The proof combines Pinkus's and Voigtlaender's universal approximation theorems with a continuity lemma and a union/intersection bound over finitely many sampled basis functions.","tokens_in":5580,"tokens_out":5663,"duration_ms":63382,"significance":"The underlying idea—that dense support of the sampling distribution, rather than absolute continuity, is enough to inherit universal approximation with high probability, together with an explicit width bound—is interesting and would be a useful contribution to the random basis expansion / random feature literature, especially for spike-and-slab type samplers. The manuscript is not constructive and relies on standard external UATs, but that is acceptable for an existence-style result. However, the central theorem as stated is mathematically ill-posed, the invocation of Voigtlaender's theorem is invalid, and the displayed sample-size bound is algebraically wrong. A correct version would likely require substantial reformulation, so the present manuscript is not suitable for publication.","major_comments":[{"comment":"The theorem is ill-posed for complex inputs. It takes f:C^n→C, K⊂C^n, but defines φ_j(x)=φ(ω_j^T x+b_j) with φ:R→C and (ω_j,b_j)∈R^{n+1}. For any x with nonzero imaginary part, ω_j^T x+b_j is not a real number, so φ_j(x) is undefined. Lemma 2.4, the only continuity mechanism, is restricted to K⊂R^n. Consequently, the object whose L∞ norm appears in (7) does not exist for general compact K⊂C^n.","section":"Section 3, Theorem 3.1 (definition of φ_j)"},{"comment":"The application of Voigtlaender's theorem is invalid. Theorem 2.3 requires φ:C→C and, through Definition 2.1, approximates continuous functions on R^n, not on C^n. The proof's sentence 'If f:C^n→C and φ:R→C, this follows from Theorem 2.3' has no support in the stated theorem. Thus Eq. (8), which supplies the approximating q-term expansion, has no valid hypothesis for the complex-input case advertised in Theorem 3.1.","section":"Section 2, Theorem 2.3 and Section 3, Eq. (8)"},{"comment":"The probability inequality has the wrong direction. Eq. (10) requires (1-(1-v_*)^l)^q ≥ 1-η. With m=ql, the l=m/q substitution gives (1-(1-v_*)^{m/q})^q ≥ 1-η, not ≤ as written in Eq. (11). Consequently Eq. (12) does not follow. In addition, the m-bound displayed in Theorem 3.1 contains an undefined constant C in the denominator and a log expression that is not derived from v_*; the statement's bound is therefore not a valid consequence of the argument.","section":"Section 3, Eqs. (10)-(12)"},{"comment":"The proof assumes that dense support implies v_i = P_D(B_{δ_i}(ω̃_i,b̃_i)) > 0 for every δ_i>0. This is true under the standard definition of support of a probability measure (every neighborhood of every support point has positive measure), but that definition is not given. If 'dense support' is interpreted only as the topological support being dense, the positivity of v_i is not automatic. This is a load-bearing point for the probabilistic statement and should be stated explicitly.","section":"Section 3, proof of Theorem 3.1 (dense support and v_i>0)"}],"minor_comments":[{"comment":"The proof refers to 'the compact set T⊂C bounded by M' but T is never defined; since φ:R→C, the relevant compact set should be a subset of R containing the values ω^T x+b. Also Eq. (6) has a typo: |ω_j^T x_1+b-(ω_j^T x_2+b)| should equal |ω_j^T(x_1-x_2)|, and the displayed line is missing a closing parenthesis and writes x for x_2.","section":"Section 2, Lemma 2.4 proof"},{"comment":"Sums over the q approximating terms are indexed with n in several places (e.g., 'nX_{i=1}' and 'Let {j_i}_{i=1}^n'); these should be q. This is confusing because n is already the input dimension.","section":"Section 3, proof of Theorem 3.1, Eq. (13) and following"},{"comment":"The phrase 'non-continuous weight distribution' is ambiguous; it could mean 'not continuous' in the topological sense or 'not absolutely continuous'. The paper should specify 'not having a density' or 'not absolutely continuous' where that is the intended meaning. Also, the abstract contains the typo 'jusify' and the introduction has 'the approximation' -> 'approximates' or similar.","section":"Throughout"},{"comment":"The logarithmic bound as typeset is malformed: 'm≥log( 1/(1-(1-η)^{1/q}) q - 1 /(1-C) )' lacks parentheses and a log base, and the dependence on C versus v_* is not explained. Please rewrite with a clearly derived expression.","section":"Section 3, Theorem 3.1 statement"}],"recommendation":"reject","confidential_remarks":"The paper's core defect is not a local typo or missing citation: the main theorem is undefined for complex inputs. A salvageable version likely exists for real inputs (f:R^n→C with φ:R→C), and the high-probability mechanism with dense support is a reasonable idea. But the current manuscript would need a substantially rewritten theorem statement, proof, and sample-size bound, and the relationship with Voigtlaender's theorem would need to be reworked. Given the advertised result is not proved as stated, rejection is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one thing to know: the paper points at a real gap — universal approximation results for random basis expansions have not covered dense-support non-continuous sampling, which matters for spike-and-slab schemes — but the main theorem as written is ill-posed. φ:R→C is evaluated at ω_j^T x + b_j with x∈C^n; for non-real x the argument is not real, so the basis functions are undefined. The proof then invokes Voigtlaender's theorem, which requires φ:C→C and, as stated, approximates functions on R^n. That bridge does not hold.\n\nWhat is genuinely new and good: the transfer idea — dense support means random samples can land near the fixed weights supplied by a universal approximation theorem, and a continuity lemma converts closeness in weights into closeness in functions. Lemma 2.4 is standard and correct for real inputs. For the real setting (f:R^n→R, φ:R→R), the strategy is sound and the high-probability claim should be recoverable after fixing the quantitative errors. The paper is honest that the result is non-constructive.\n\nSoft spots beyond the domain error: the log bound in (12) is algebraically wrong, the inequality in (11) has the wrong direction, and the coefficient sum in (14) drops absolute values, so the error bound is not established even in the real case. The statement mixes 'non-polyharmonic' with a real-valued activation; for φ:R→C the term is undefined, and for φ:R→R it should read 'non-polynomial'. The complex activation case is simply not handled by the cited theorem.\n\nThis deserves a serious referee only because the real-input variant is plausibly repairable and the sampling question is relevant. As written, it should not go forward. If the authors restrict the theorem to real inputs, or replace φ:R→C with a genuinely complex activation and adjust the inputs accordingly, the resulting paper could be worth publishing.","headline":"The advertised complex case is not defined, but the real-input transfer idea is worth a repair.","tokens_in":6034,"tokens_out":3669,"would_cite":false,"duration_ms":41123,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["41A30","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"Random basis expansions with non-continuous dense-support weight sampling and complex-valued activations approximate any continuous function on compact sets with arbitrarily high probability, governed by a logarithmic sample-count bound.","keywords":["random basis expansion","universal approximation","dense support","non-continuous weight sampling","complex-valued activation","high probability","spike-and-slab sampling","random features"],"falsifier":"Take any continuous non-polyharmonic φ:R→C and any point x∈C^n with ωᵀx+b∉R; since φ is only defined on R, the basis function is undefined at that point. Placing such a point in the compact set K shows the theorem's statement cannot even be evaluated as written, so the claim fails for the stated domain.","tokens_in":5198,"feed_emoji":"🎲","tokens_out":6436,"duration_ms":67192,"temperature":0.7,"pith_summary":"The paper sets out to extend universal approximation results for random basis expansions to two untested settings: weight distributions that are not continuous but have dense support, and activation functions that are complex-valued. Its main theorem claims that if the weights and biases are sampled iid from such a distribution, then for any continuous target on a compact set and any accuracy, a random-basis expansion with a sufficiently large—and logarithmically bounded—number of samples is within that accuracy with probability at least 1−η. The practical payoff is that sparse and spike-and-slab sampling schemes, which current theory does not cover, would inherit the same density guarantee as continuous sampling, and complex-valued feature maps used in random Fourier methods would also qualify. The proof works by taking a known approximating expansion, sampling weights into small neighborhoods of its component weights, and union-bounding the chance that any neighborhood is missed.","feed_headline":"Random bases approximate any function even with sparse weight sampling","feed_subtitle":"The paper proves dense-support non-continuous sampling still yields universal approximation with high probability.","key_machinery":"The central object is the random basis expansion (1), a single-layer network with random hidden weights and tunable output coefficients. The argument's engine is Lemma 2.4, a continuity result: as weights (ω_j,b_j) converge in R^{n+1}, the induced feature functions converge in L∞ on compact sets, provided φ is continuous. This lets the proof transplant universal approximation from the fully-trained setting: for each term of a reference expansion, the probability that a sampled weight lands in a δ-ball around the reference weight is positive because the sampling distribution has dense support; a union bound over the q reference terms yields the sample-count formula m ≥ log(...). The complex c","core_discovery":"In the paper's own terms, the central discovery is Theorem 3.1: for any continuous f:C^n→C, any continuous non-polyharmonic φ:R→C, and any iid sample distribution D with dense support in R^{n+1}, for every compact K⊂C^n and ε>0 there is an m (logarithmic in 1/η) such that with probability at least 1−η the expansion Σⱼ αⱼ φ(ωⱼᵀx+bⱼ) lies within ε of f in L∞(K). The coefficients are chosen by matching sampled weights to the weights of a q-term expansion guaranteed by classical universal approximation, zeroing the rest. The theorem is an existence result—it does not construct the coefficients—and extends the universal approximation property from continuous weight distributions to any dense-supp","pith_inferences":["As stated, the complex-input case likely does not go through: the proof invokes a universal approximation theorem that demands a complex-to-complex activation and real inputs, while the theorem is stated with φ:R→C and x∈C^n, leaving φ(ωᵀx+b) undefined for non-real inputs; a corrected theorem would probably restrict inputs to R^n or take φ:C→C.","The proof silently assumes that dense support in R^{n+1} implies every δ-ball has positive D-measure; this is true under the usual support definition but is not shown, and a distribution supported on a dense countable set with atomic masses would need an explicit justification that no ball has zero mass.","A testable next step is to specialize the result to random Fourier features on real inputs, where φ(t)=e^{it} (or a suitable complex extension) and the sampling distribution is dense; verifying the constants v* and q for this case would turn the logarithmic bound into a practical sample-size guide."],"forward_implications":["Sparse, spike-and-slab weight sampling schemes gain a universal approximation guarantee with high probability, provided their support is dense in the weight space.","The high-probability guarantee becomes almost sure as the sample count grows, and the required m grows only logarithmically in 1/η.","Complex-valued activations (e.g., the complex exponentials used in random Fourier features) are covered by the theorem's statement, subject to the activation satisfying the non-polyharmonic condition.","The theorem is non-constructive: it certifies existence of approximating expansions and coefficient values but does not provide an algorithm to find them."],"fun_headline_variants":["Non-continuous weight draws? Still universal approximation","Dense support in non-continuous sampling: universal approximation","High-probability universal approximation from non-continuous weights","Random bases approximate any function with non-continuous weights","Universal approximation holds for random bases with dense-support sampling"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof's load-bearing premise is that a known universal approximation theorem for complex-valued activations can be invoked when the activation is real-to-complex and the input is allowed to be complex, even though the relevant feature expression is undefined for non-real inputs; a secondary unproven premise is that dense support forces every small ball to have positive sampling probability.","fun_headline_variants_meta":{"raw":{"variants":["Non-continuous weight draws? Still universal approximation","Dense support in non-continuous sampling: universal approximation","High-probability universal approximation from non-continuous weights","Random bases approximate any function with non-continuous weights","Universal approximation holds for random bases with dense-support sampling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001684,"raw_usage":{"total_tokens":6463,"prompt_tokens":645,"completion_tokens":5818,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":389,"completion_tokens_details":{"reasoning_tokens":5741}},"tokens_in":389,"tokens_out":5818,"duration_ms":40465,"temperature":1.0,"reasoning_tokens":5741,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T20:38:22.855471+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any continuous non-polyharmonic φ:R→C and any point x∈C^n with ωᵀx+b∉R; since φ is only defined on R, the basis function is undefined at that point. Placing such a point in the compact set K shows the theorem's statement cannot even be evaluated as written, so the claim fails for the stated domain.","supporting_citations":[],"review_version":1}