Acta Mathematicae Applicatae Sinica, English Series最新文献

Tensor data have been widely used in many fields, e.g., modern biomedical imaging, chemometrics, and economics, but often suffer from some common issues as in high dimensional statistics. How to find their low-dimensional latent structure has been of great interest for statisticians. To this end, we develop two efficient tensor sufficient dimension reduction methods based on the sliced average variance estimation (SAVE) to estimate the corresponding dimension reduction subspaces. The first one, entitled tensor sliced average variance estimation (TSAVE), works well when the response is discrete or takes finite values, but is not (sqrt n) consistent for continuous response; the second one, named bias-correction tensor sliced average variance estimation (CTSAVE), is a de-biased version of the TSAVE method. The asymptotic properties of both methods are derived under mild conditions. Simulations and real data examples are also provided to show the superiority of the efficiency of the developed methods.

Let F, G and H be three graphs with G ⊆ H. We call G an F-saturated graph relative to H, if there is no copy of F in G but there is a copy of F in G + e for any e ∈ E(H) E(G). The F-saturation game on host graph H consists of two players, named Max and Min, who alternately add edges of H to G such that each chosen edge avoids creating a copy of F in G, and the players continue to choose edges until G becomes F-saturated relative to H. Max wishes to maximize the length of the game, while Min wishes to minimize the process. Let sat_g(F, H) (resp. sat′_g(F, H)) denote the number of edges chosen when Max (resp. when Min) starts the game and both players play optimally. In this article, we show that sat_g(P₅, K_n) = sat′_g(P₅, K_n) = n + 2 for n ≥ 15, and sat_g(P₅, K_m,n), sat′_g(P₅, K_m,n) lie in (left{ {m + n - leftlfloor {{{m - 2} over 4}} rightrfloor ,,m + n - leftlceil {{{m - 3} over 4}} rightrceil } right}) if (n ge {5 over 2}m) and m ≥ 4, respectively.

This paper aims to construct a two-grid scheme of fully discretized expanded mixed finite element methods for optimal control problems governed by parabolic integro-differential equations and discuss a priori error estimates. The state variables and co-state variables are discretized by the lowest order Raviart-Thomas mixed finite element, and the control variable is approximated by piecewise constant functions. The time derivative is discretized by the backward Euler method. Firstly, we define some new mixed elliptic projections and prove the corresponding error estimates which play an important role in subsequent convergence analysis. Secondly, we derive a priori error estimates for all variables. Thirdly, we present a two-grid scheme and analyze its convergence. In the two-grid scheme, the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid and the solution of a decoupled linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. At last, a numerical example is presented to verify the theoretical results.

In this paper, we propose a new method for spread option pricing under the multivariate irreducible diffusions without jumps and with different types of jumps by the expansion of the transition density function. By the quasi-Lamperti transform, which unitizes the diffusion matrix at the initial time, and applying the small-time Itô-Taylor expansion method, we derive explicit recursive formulas for the expansion coefficients of transition densities and spread option prices for multivariate diffusions with jumps in return. It is worth mentioning that we also give the closed-form formula of spread option price whose underlying asset price processes contain a Merton jump and a double exponential jump, which is innovative compared with current literature. The theoretical proof of convergence is presented in detail.

In this paper, we design the differentially private variants of the classical Frank-Wolfe algorithm with shuffle model in the optimization of machine learning. Under weak assumptions and the generalized linear loss (GLL) structure, we propose a noisy Frank-Wolfe with shuffle model algorithm (NoisyFWS) and a noisy variance-reduced Frank-Wolfe with the shuffle model algorithm (NoisyVRFWS) by adding calibrated laplace noise under shuffling scheme in the ℓ_p(p ∈ [1, 2])-case, and study their privacy as well as utility guarantees for the Hölder smoothness GLL. In particular, the privacy guarantees are mainly achieved by using advanced composition and privacy amplification by shuffling. The utility bounds of the NoisyFWS and NoisyVRFWS are analyzed and obtained the optimal excess population risks ({cal O}({n^{ - {{1 + alpha } over {4alpha }}}} + {{log (d)sqrt {log ({1 mathord{left/ {vphantom {1 delta }} right.} delta })} } over {nepsilon,}})) and ({cal O}({n^{ - {{1 + alpha } over {4alpha }}}} + {{log (d)sqrt {log ({1 mathord{left/ {vphantom {1 delta }} right.} delta })} } over {{n^2epsilon},}})) with gradient complexity ({cal O}({n^{ - {{{{(1 + alpha )}^2}} over {4{alpha ^2}}}}})) for (alpha in left[ {{1 mathord{left/ {vphantom {1 {sqrt 3 ,,1}}} right.} {sqrt 3 ,,1}}} right]). It turns out that the risk rates under shuffling scheme are a nearly-dimension independent rate, which is consistent with the previous work in some cases. In addition, there is a vital tradeoff between (α, L)-Hölder smoothness GLL and the gradient complexity. The linear gradient complexity ({cal O}(n)) is showed by the parameter α = 1.