The purpose of this paper is to solve the seventh-order functional equation as follows:  --------------------------- Next, we study the stability of this type of functional equation. Clearly, the function ----------  holds in this type functional equation. Also, we prove Hyers-Ulam stability for this type functional equation in the β-Gaussian Banach space.  

An unknown m by n matrix X0 is to be estimated from noisy measurements Y = X0 + Z, where the noise matrix Z has i.i.d Gaussian entries. A popular matrix denoising scheme solves the nuclear norm penalization problem minX‖Y − X‖F /2 + λ‖X‖∗, where ‖X‖∗ denotes the nuclear norm (sum of singular values). This is the analog, for matrices, of `1 penalization in the vector case. It has been empiricall...

Let X0 be an unknown M by N matrix. In matrix recovery, one takes n < MN linear measurements y1, . . . , yn of X0, where yi = Tr(ai X0) and each ai is a M by N matrix. For measurement matrices with Gaussian i.i.d entries, it known that if X0 is of low rank, it is recoverable from just a few measurements. A popular approach for matrix recovery is Nuclear Norm Minimization (NNM): solving the conv...

Let X(0) be an unknown M by N matrix. In matrix recovery, one takes n < MN linear measurements y(1),…,y(n) of X(0), where y(i) = Tr(A(T)iX(0)) and each A(i) is an M by N matrix. A popular approach for matrix recovery is nuclear norm minimization (NNM): solving the convex optimization problem min ||X||*subject to y(i) =Tr(A(T)(i)X) for all 1 ≤ i ≤ n, where || · ||* denotes the nuclear norm, name...

We develop the exact constant of the risk asymptotics in the uniform norm for density estimation. This constant has first been found for nonparametric regression and for signal estimation in Gaussian white noise. Hölder classes for arbitrary smoothness index β > 0 on the unit interval are considered. The constant involves the value of an optimal recovery problem as in the white noise case, but ...

Consider the estimation of a signal x ∈ R from noisy observations r = x+ z, where the input x is generated by an independent and identically distributed (i.i.d.) Gaussian mixture source, and z is additive white Gaussian noise (AWGN) in parallel Gaussian channels. Typically, the `2-norm error (squared error) is used to quantify the performance of the estimation process. In contrast, we consider ...

Since the work of Gauss, number theorists have been interested in analogues of Z where concepts from arithmetic can also be developed. The example we will look at in this handout is the Gaussian integers: Z[i] = {a + bi : a, b ∈ Z}. Excluding the last two sections of the handout, the topics we will study are extensions of common properties of the integers. Here is what we will cover in each sec...

We consider the following sparse signal recovery (or feature selection) problem: given a design matrix X ∈ Rn×m (m ≫ n) and a noisy observation vector y ∈ Rn satisfying y = Xβ∗ + ε where ε is the noise vector following a Gaussian distribution N(0,σ2I), how to recover the signal (or parameter vector) β∗ when the signal is sparse? The Dantzig selector has been proposed for sparse signal recovery ...

Let {Rt, 0 ≤ t ≤ 1} be a symmetric α-stable Riemann-Liouville process with Hurst parameterH > 0. Consider a translation invariant, β-self-similar, and p-pseudo-additive functional semi-norm ||.||. We show that if H > β+1/p and γ = (H − β− 1/p), then lim ε↓0 ε logP [||R|| ≤ ε] = −K ∈ [−∞, 0), with K finite in the Gaussian case α = 2. If α < 2, we prove that K is finite when R is continuous and H...

Consider the estimation of a signal x ∈ R N from noisy observations r = x + z, where the input x is generated by an independent and identically distributed (i.i.d.) Gaussian mixture source, and z is additive white Gaussian noise (AWGN) in parallel Gaussian channels. Typically, the ℓ2-norm error (squared error) is used to quantify the performance of the estimation process. In contrast, we consid...