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- research-articleJune 2024
Learning Quantum Hamiltonians at Any Temperature in Polynomial Time
STOC 2024: Proceedings of the 56th Annual ACM Symposium on Theory of ComputingJune 2024, Pages 1470–1477https://doi.org/10.1145/3618260.3649619We study the problem of learning a local quantum Hamiltonian H given copies of its Gibbs state ρ = e−β H/(e−β H) at a known inverse temperature β>0. Anshu, Arunachalam, Kuwahara, and Soleimanifar gave an algorithm to learn a Hamiltonian on n qubits to ...
- research-articleSeptember 2022
Optimal-degree polynomial approximations for exponentials and gaussian kernel density estimation
CCC '22: Proceedings of the 37th Computational Complexity ConferenceJuly 2022, Article No.: 22, Pages 1–23https://doi.org/10.4230/LIPIcs.CCC.2022.22For any real numbers B ≥ 1 and δ ∈ (0, 1) and function f : [0, B] → R, let dB;δ(f)∈Z>0 denote the minimum degree of a polynomial p(x) satisfying supx∈[0,B]|p(x)−f(x)|<δ. In this paper, we provide precise asymptotics for dB;δ(e)−x) and dB;δ(e)x) in terms ...
- research-articleJune 2022
The approximate degree of DNF and CNF formulas
STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of ComputingJune 2022, Pages 1194–1207https://doi.org/10.1145/3519935.3520000The approximate degree of a Boolean function f∶{0,1}n→{0,1} is the minimum degree of a real polynomial p that approximates f pointwise: |f(x)−p(x)|≤1/3 for all x∈{0,1}n. For any δ>0, we construct DNF and CNF formulas of polynomial size with approximate ...
- research-articleMarch 2022
- research-articleSeptember 2021
Improved Bounds on Fourier Entropy and Min-entropy
ACM Transactions on Computation Theory (TOCT), Volume 13, Issue 4Article No.: 22, Pages 1–40https://doi.org/10.1145/3470860Given a Boolean function f:{ -1,1} ^{n}→ { -1,1, define the Fourier distribution to be the distribution on subsets of [n], where each S ⊆ [n] is sampled with probability f ˆ (S)2. The Fourier Entropy-influence (FEI) conjecture of Friedgut and Kalai [28] ...
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- research-articleJanuary 2021
Lower Bounding the AND-OR Tree via Symmetrization
ACM Transactions on Computation Theory (TOCT), Volume 13, Issue 1Article No.: 3, Pages 1–11https://doi.org/10.1145/3434385We prove a simple, nearly tight lower bound on the approximate degree of the two-level AND-OR tree using symmetrization arguments. Specifically, we show that ˜deg(ANDm ˆ ORn) = ˜Ω(√mn). We prove this lower bound via reduction to the OR function through a ...
- research-articleJanuary 2021
Optimal feedback law recovery by gradient-augmented sparse polynomial regression
The Journal of Machine Learning Research (JMLR), Volume 22, Issue 1Article No.: 48, Pages 2205–2236A sparse regression approach for the computation of high-dimensional optimal feedback laws arising in deterministic nonlinear control is proposed. The approach exploits the control-theoretical link between Hamilton-Jacobi-Bellman PDEs characterizing the ...
- research-articleJanuary 2021
Lasso Hyperinterpolation Over General Regions
SIAM Journal on Scientific Computing (SISC), Volume 43, Issue 62021, Pages A3967–A3991https://doi.org/10.1137/20M137793XThis paper develops a fully discrete soft thresholding polynomial approximation over a general region, named Lasso hyperinterpolation. This approximation is an $\ell_1$-regularized discrete least squares approximation under the same conditions of ...
- research-articleJanuary 2021
Deep Network Approximation for Smooth Functions
SIAM Journal on Mathematical Analysis (SIMA), Volume 53, Issue 52021, Pages 5465–5506https://doi.org/10.1137/20M134695XThis paper establishes the optimal approximation error characterization of deep rectified linear unit (ReLU) networks for smooth functions in terms of both width and depth simultaneously. To that end, we first prove that multivariate polynomials can be ...
- research-articleJanuary 2021
Error Localization of Best $L_{1}$ Polynomial Approximants
SIAM Journal on Numerical Analysis (SINUM), Volume 59, Issue 12021, Pages 314–333https://doi.org/10.1137/19M1242860An important observation in compressed sensing is that the $\ell_0$ minimizer of an underdetermined linear system is equal to the $\ell_1$ minimizer when there exists a sparse solution vector and a certain restricted isometry property holds. Here, we ...
- research-articleNovember 2020
Novel hardware & software design for mathematical and AI acceleration
CASCON '20: Proceedings of the 30th Annual International Conference on Computer Science and Software EngineeringNovember 2020, Pages 268–269Special-purpose computational hardware, in integer and floating-point arithmetic units, as well as in memory systems, provides opportunities to accelerate a wide range of mathematical applications, including medical imaging, digital signal processing, ...
- research-articleJanuary 2020
On Separating Points for Ensemble Controllability
SIAM Journal on Control and Optimization (SICON), Volume 58, Issue 52020, Pages 2740–2764https://doi.org/10.1137/19M1278648Recent years have witnessed a wave of research activities in systems science toward the study of population systems. The driving force behind this shift was geared by numerous emerging and ever-changing technologies in life and physical sciences and ...
- research-articleJanuary 2020
An Optimal Polynomial Approximation of Brownian Motion
SIAM Journal on Numerical Analysis (SINUM), Volume 58, Issue 32020, Pages 1393–1421https://doi.org/10.1137/19M1261912In this paper, we will present a strong (or pathwise) approximation of standard Brownian motion by a class of orthogonal polynomials. The coefficients that are obtained from the expansion of Brownian motion in this polynomial basis are independent Gaussian ...
- research-articleJanuary 2019
A Nearly Optimal Lower Bound on the Approximate Degree of AC$^0$
SIAM Journal on Computing (SICOMP), Volume 49, Issue 42020, Pages FOCS17-59–FOCS17-96https://doi.org/10.1137/17M1161737The approximate degree of a Boolean function $f \colon \{-1, 1\}^n \rightarrow \{-1, 1\}$ is the least degree of a real polynomial that approximates $f$ pointwise to error at most 1/3. We introduce a generic method for increasing the approximate degree of ...
- research-articleJuly 2018
What about interpolation?: a radial basis function approach to classifier prediction modeling in XCSF
GECCO '18: Proceedings of the Genetic and Evolutionary Computation ConferenceJuly 2018, Pages 537–544https://doi.org/10.1145/3205455.3205599Learning Classifier Systems (LCS) have been strongly investigated in the context of regression tasks and great successes have been achieved by applying the function approximating Extended Classifier System (XCSF) endowed with sophisticated prediction ...
- research-articleMarch 2018
Cryptanalysis of Morillo–Obrador polynomial delegation schemes
IET Information Security (ISE2), Volume 12, Issue 2March 2018, Pages 127–132https://doi.org/10.1049/iet-ifs.2017.0259Verifiable computation (VC) allows a client to outsource (delegate) the computation of a function f on an input x to a server and then verify the server's results with substantially less time than computing f (x) from scratch. The security of VC requires ...
- research-articleJanuary 2018
Numerical Integration in Multiple Dimensions with Designed Quadrature
SIAM Journal on Scientific Computing (SISC), Volume 40, Issue 42018, Pages A2033–A2061https://doi.org/10.1137/17M1137875We present a systematic computational framework for generating positive quadrature rules in multiple dimensions on general geometries. A direct moment-matching formulation that enforces exact integration on polynomial subspaces yields nonlinear conditions ...
- research-articleJanuary 2018
Polynomial Approximation of High-Dimensional Hamilton--Jacobi--Bellman Equations and Applications to Feedback Control of Semilinear Parabolic PDEs
SIAM Journal on Scientific Computing (SISC), Volume 40, Issue 22018, Pages A629–A652https://doi.org/10.1137/17M1116635A procedure for the numerical approximation of high-dimensional Hamilton--Jacobi--Bellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed. Its main ingredients are a pseudospectral collocation ...
- research-articleJanuary 2018
The Power of Asymmetry in Constant-Depth Circuits
SIAM Journal on Computing (SICOMP), Volume 47, Issue 62018, Pages 2362–2434https://doi.org/10.1137/16M1064477The threshold degree of a Boolean function $f$ is the minimum degree of a real polynomial $p$ that represents $f$ in sign: $f(x)\equiv {sgn}~ p(x)$. Introduced in the seminal work of Minsky and Papert [Perceptrons: An Introduction to Computational Geometry, ...
- research-articleDecember 2015
Complexity‐aware‐normalised mean squared error ‘CAN’ metric for dimension estimation of memory polynomial‐based power amplifiers behavioural models
IET Communications (CMU2), Volume 9, Issue 18Pages 2227–2233https://doi.org/10.1049/iet-com.2015.0371The memory polynomial model is widely used for the behavioural modelling of radio‐frequency non‐linear power amplifiers having memory effects. One challenging task related to this model is the selection of its dimension which is defined by the non‐...