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Sunghyeon Jo

Publications and source records attributed to Sunghyeon Jo.

9 recordsLinked to original sources
arXiv2026

On Minimax Optimality and Uniqueness of Fixed-Step First-Order Methods for Smooth Convex Optimization

This paper considers the design of optimal fixed-step first-order methods for high-dimensional minimization of $L$-smooth convex functions. For optimizing worst-case performance measured via suboptimality of the final function value (relative to the initial squared distance to a minimizer), we provide an algebraic proof of the optimality of the optimized gradient method (OGM) and establish its uniqueness among all fixed-step first-order methods. For the alternative measure of final squared gradient norm (relative to initial suboptimality), we prove the OGM-G method is optimal and uniquely so among fixed-step first-order methods. Finally, for the setting measuring the final squared gradient norm (relative to the initial squared distance to a minimizer), we show the recently proposed Lemniscate method is optimal and uniquely so. Our proofs rely on algebraic reductions for lower bound arguments rather than traditional information-theoretic bounds, which were previously only able to establish OGM's optimality but not uniqueness.

math.OC↗
arXiv2026

An Improved Upper Bound for Multicolour Ramsey Numbers

Let $R_r(k)$ denote the diagonal $r$-colour Ramsey number. We prove that there exist absolute constants $c,K>0$ such that $R_r(k)\le r^{rk}\exp\!\left(-c\frac{k}{r\log^2(2r)}\right)$ for every $r\ge2$ and every $k\ge Kr^2\log^6(2r)$. This improves the exponential saving in a recent bound of Yang and Mao by a factor of order $r\log^2(2r)$. The proof proceeds through an off-diagonal bound, which asymptotically improves the classical multinomial bound throughout a neighbourhood of the diagonal.

math.CO↗
arXiv2026

The Exact Online Threshold for the Asymmetric Binary Perceptron

Let $G\in\mathbb{R}^{M\times N}$ have independent standard Gaussian entries. For a fixed margin $κ\in\mathbb{R}$, the asymmetric binary perceptron asks for $σ\in\{\pm1\}^N$ such that $Gσ/\sqrt{N}\geκ\mathbf{1}_M$. We study the online version of this problem, in which the columns of $G$ arrive sequentially and each sign must be chosen irrevocably before future columns are revealed. We determine the exact threshold $α_{\mathrm{on}}(κ)$ for every fixed $κ$: for $M/N\toα$ with $α<α_{\mathrm{on}}(κ)$, there is a deterministic online algorithm, using $O(MN)$ arithmetic operations and polynomial bit complexity, that succeeds with high probability, while for $α>α_{\mathrm{on}}(κ)$, no online algorithm succeeds with high probability. The threshold is characterized by a one-dimensional stochastic control problem for Brownian motion. The main difficulty is to upgrade a single-coordinate Brownian limit to simultaneous feasibility of all $M=Θ(N)$ constraints, which we do with half-line monotonicity and a short final correction block. At zero margin, we give a computer-assisted proof that $0.32747<α_{\mathrm{on}}(0)<0.36664$. In particular, every density below $0.32747$ is achievable online by such an algorithm, more than tripling the best density previously proved attainable by any polynomial-time algorithm, online or offline (the previous bound was $α\le0.1$, due to Li, Schramm, and Zhou). As $κ\to+\infty$, the online threshold agrees to first order with the offline storage capacity. As $κ\to-\infty$, it has the same asymptotic scale as the best known offline polynomial-time guarantee, while the storage capacity is larger by a factor of order $κ^2$.

cs.DS↗
arXiv2026

Shor's Conjecture Is True: Projective Measurements Suffice for Binary Accessible Information

Shor conjectured that a von Neumann measurement attains the accessible information of every binary quantum ensemble. We prove the conjecture constructively in arbitrary finite dimension. For every finite-outcome positive operator-valued measure (POVM) $M$, we form an operator $T_M$ from the posterior label probabilities and show that its spectral projection-valued measure (PVM) $Π_M$ satisfies $I_{Π_M}(X{:}Y)\ge I_M(X{:}Y)$; every rank-one refinement retains the inequality. Two applications of Jensen's operator inequality prove the comparison and yield an exact concave variational formula for the accessible information. The result is a special case of the general theorem of Fang, Fawzi, and Fawzi on measured $f$-divergences; the proof below isolates the binary argument and makes the replacement $M\mapsto T_M\mapstoΠ_M$ explicit.

quant-ph↗
arXiv2026

The Sample Complexity of Fidelity Estimation to a Known Rank-$r$ Reference State Is $\widetildeΘ(r^2/\varepsilon^2)$

We settle the sample complexity of estimating the root Uhlmann fidelity $F(ρ,σ)=\operatorname{tr}\sqrt{\sqrtσρ\sqrtσ}$ between an unknown state $ρ$ and a known rank-$r$ reference state $σ$. Writing $S(r,\varepsilon)$ for the sample complexity at additive error $\varepsilon$, we resolve the open problem posed by Wang by closing, up to logarithmic factors, the gap between the previously known bounds $Ω(r/\varepsilon^2)$ and $O(r^2/\varepsilon^2)$. We prove $S(r,\varepsilon)=\widetildeΘ(r^2/\varepsilon^2)$ for all $0<\varepsilon\le\varepsilon_0$, where $\varepsilon_0>0$ is a universal constant. The lower bound already holds on a $2r$-dimensional system when $σ$ is maximally mixed on a fixed $r$-dimensional subspace, and for a hard family of states that do not commute with $σ$. The proof combines exact spectral moment matching, a radially size-biased doubly correlated Wishart model, and the Cauchy identity, reducing state indistinguishability to a long-cycle estimate for a weighted random permutation. A direct-sum embedding and binomial thinning yield the optimal $1/\varepsilon^2$ dependence. We also prove a near-quadratic lower bound $\widetildeΩ(r^2)$ for quantum spectrum estimation at constant accuracy. Combined with the recent $O(r^2(\log\log r/\log r)^2)$ upper bound, this determines the polynomial order of the sample complexity in this regime and establishes a near-quadratic barrier.

quant-ph↗
arXiv2026

Efficient Exact Quantum Sampling from the Sun-Wootters Distribution for Optimal Polynomial Intersection

Optimal Polynomial Intersection (OPI) is a structured optimization problem for which Decoded Quantum Interferometry (DQI) attains a satisfaction guarantee governed by the semicircle law. Sun and Wootters recently showed that, for balanced OPI over prime fields, a Fourier-defined distribution $P_u$ gives a strict worst-case improvement from limiting rate $0.6225$ onward and asymptotically perfect solutions from rate $0.7496$ onward, and asked whether $P_u$ can be sampled efficiently. We answer this question for Reed--Solomon OPI parameters satisfying their exponent condition strictly below the dual Johnson radius. Under coherent membership-oracle access, we give a bounded-error polynomial-time quantum sampler for $P_u$. The ideal circuit samples $P_u$ exactly conditioned on success, while a finite-precision implementation achieves any prescribed inverse-polynomial total-variation error. Consequently, every fixed limiting rate $0.6225\le r<1$ admits a strict worst-case improvement over the DQI semicircle value, and every limiting rate $r\ge 3/4$ admits solutions of satisfaction $1-o(1)$ with high probability. The algorithm coherently sums the amplitudes of all low-weight errors in each syndrome class using deterministic complete list decoding. Complete Reed--Solomon list decoding and the Sun--Wootters denominator estimate make the list size and postselection overhead polynomial. In concurrent and independent work, Horinaga and Yamakawa obtain worst-case OPI algorithms over prime-power fields and exact satisfaction at every fixed rate strictly above $3/4$.

quant-ph↗
arXiv2026

Finding a Solution to the Erdős-Ginzburg-Ziv Theorem in Linear Time

The Erdős-Ginzburg-Ziv theorem states that every sequence of 2n - 1 integers contains a subsequence of length n whose sum is divisible by n. Choi, Kang, and Lim gave a simple deterministic O(n log n) algorithm for finding such a subsequence, and Leung recently improved this to O(n log log log n). We give a deterministic linear-time algorithm. The core is a linear-time algorithm for the following prime target subset-sum problem: given p - 1 nonzero residues in Z_p and a target residue, find a subset with the prescribed sum. Our algorithm maintains a compact arithmetic-progression representation of reachable sums. When two progressions intersect, a bounded Frobenius interval in their sum allows them to be merged into one longer progression, with enough growth to pay for the update. When the representation either contains a full progression or covers all nonzero residues, the target residue is recovered constructively. The standard multiplicative reduction then extends the prime algorithm to arbitrary moduli.

cs.DS↗
arXiv2026

Tighter Bounds for the Random-Offerer Mechanism in Bilateral Trade

The random-offerer mechanism for bilateral trade selects the seller or the buyer uniformly and lets the selected agent make a profit-maximizing take-it-or-leave-it offer. Let $ρ_{\rm RO}$ be the infimum, over independent value distributions, of the mechanism's gains from trade divided by first-best gains from trade. We prove $\frac1π\le ρ_{\rm RO}<0.460242308085529$. For the lower bound, we improve the previous guarantee from approximately $0.317844$ to $1/π\approx 0.318310$. The proof uses a parameterized Lagrangian bound for pointwise-monotone allocations. At multiplier one, this bound has coefficient $2/π$, and the Lagrangian separates into two terms controlled by the optimal seller-offering and buyer-offering profits. For the upper bound, we construct an explicit family consisting of a truncated equal-revenue buyer and a seller distribution with a tilted power-law lower tail and a constant-virtual-cost segment. The family satisfies $\operatorname{FB}/\operatorname{RO}>2.17276852308451$, improving the previous explicit ratio $2.0749$; rigorous interval arithmetic certifies the numerical inequality.

cs.GT↗