Sparse Hopfield network reconstruction with ℓ1\ell_{1} regularization

We propose an efficient strategy to infer sparse Hopfield network based on magnetizations and pairwise correlations measured through Glauber samplings. This strategy incorporates the $\ell_{1}$ regularization into the Bethe approximation by a quadratic approximation to the log-likelihood, and is able to further reduce the inference error of the Bethe approximation without the regularization. The optimal regularization parameter is observed to be of the order of $M^{-\nu}$ where $M$ is the number of independent samples. The value of the scaling exponent depends on the performance measure. $\nu\simeq0.5001$ for root mean squared error measure while $\nu\simeq0.2743$ for misclassification rate measure. The efficiency of this strategy is demonstrated for the sparse Hopfield model, but the method is generally applicable to other diluted mean field models. In particular, it is simple in implementation without heavy computational cost.Comment: 9 pages, 3 figures, Eur. Phys. J. B (in press

Speculation in Standard Auctions with Resale

According to the theory of incomplete contracts, given nonverifiable entrepreneurial project choices together with divergent objectives between an entrepreneur and its outside financier, the entrepreneur can credibly pledge only part of its project outcome for external funding. Meanwhile, entrepreneurial net worth must be put as down payment to ameliorate agency costs. In a real dynamic general equilibrium model with heterogeneous agents and nonverifiable project choices, endogenous agency costs significantly change the businesscycle pattern in the sense that the model can replicate an important empirical fact, the amplified hump-shaped output behavior. Furthermore, variable asset prices can a ect entrepreneurial net worth and then subsequently change the dynamic features of aggregate output along business cycles.Asset Prices, Business Cycles, Credit Constraints, Hump-Shaped Output Dynamics, Nonverifiable Project Choice

Double Shuffle Relations of Double Zeta Values and Double Eisenstein Series of Level N

In their seminal paper "Double zeta values and modular forms" Gangl, Kaneko and Zagier defined a double Eisenstein series and used it to study the relations between double zeta values. One of their key ideas is to study the formal double space and apply the double shuffle relations. They also proved the double shuffle relations for the double Eisenstein series. More recently, Kaneko and Tasaka extended the double Eisenstein series to level 2, proved its double shuffle relations and studied the double zeta values of level 2. Motivated by the above works, we define in this paper the corresponding objects at higher levels and prove that the double Eisenstein series of level N satisfies the double shuffle relations for every positive integer N. In order to obtain our main theorem we prove a key result on the multiple divisor functions of level N and then use it to solve a complicated under-determined system of linear equations by some standard techniques from linear algebra.Comment: A new section 5 is added and a new Theorem 6.5 is prove