Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNew approach to Bayesian high-dimensional linear regression
178
0
0.0
(
0
)
Added by
Shirin Jalali
Publication date
2016
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Consider the problem of estimating parameters $X^n in mathbb{R}^n $, generated by a stationary process, from $m$ response variables $Y^m = AX^n+Z^m$, under the assumption that the distribution of $X^n$ is known. This is the most general version of the Bayesian linear regression problem. The lack of computationally feasible algorithms that can employ generic prior distributions and provide a good estimate of $X^n$ has limited the set of distributions researchers use to model the data. In this paper, a new scheme called Q-MAP is proposed. The new method has the following properties: (i) It has similarities to the popular MAP estimation under the noiseless setting. (ii) In the noiseless setting, it achieves the asymptotically optimal performance when $X^n$ has independent and identically distributed components. (iii) It scales favorably with the dimensions of the problem and therefore is applicable to high-dimensional setups. (iv) The solution of the Q-MAP optimization can be found via a proposed iterative algorithm which is provably robust to the error (noise) in the response variables.
rate research
Read More
In this paper study the problem of signal detection in Gaussian noise in a distributed setting. We derive a lower bound on the size that the signal needs to have in order to be detectable. Moreover, we exhibit optimal distributed testing strategies that attain the lower bound.
We study minimization of a parametric family of relative entropies, termed relative $alpha$-entropies (denoted $mathscr{I}_{alpha}(P,Q)$). These arise as redundancies under mismatched compression when cumulants of compressed lengths are considered instead of expected compressed lengths. These parametric relative entropies are a generalization of the usual relative entropy (Kullback-Leibler divergence). Just like relative entropy, these relative $alpha$-entropies behave like squared Euclidean distance and satisfy the Pythagorean property. Minimization of $mathscr{I}_{alpha}(P,Q)$ over the first argument on a set of probability distributions that constitutes a linear family is studied. Such a minimization generalizes the maximum R{e}nyi or Tsallis entropy principle. The minimizing probability distribution (termed $mathscr{I}_{alpha}$-projection) for a linear family is shown to have a power-law.
As in standard linear regression, in truncated linear regression, we are given access to observations $(A_i, y_i)_i$ whose dependent variable equals $y_i= A_i^{rm T} cdot x^* + eta_i$, where $x^*$ is some fixed unknown vector of interest and $eta_i$ is independent noise; except we are only given an observation if its dependent variable $y_i$ lies in some truncation set $S subset mathbb{R}$. The goal is to recover $x^*$ under some favorable conditions on the $A_i$s and the noise distribution. We prove that there exists a computationally and statistically efficient method for recovering $k$-sparse $n$-dimensional vectors $x^*$ from $m$ truncated samples, which attains an optimal $ell_2$ reconstruction error of $O(sqrt{(k log n)/m})$. As a corollary, our guarantees imply a computationally efficient and information-theoretically optimal algorithm for compressed sensing with truncation, which may arise from measurement saturation effects. Our result follows from a statistical and computational analysis of the Stochastic Gradient Descent (SGD) algorithm for solving a natural adaptation of the LASSO optimization problem that accommodates truncation. This generalizes the works of both: (1) [Daskalakis et al. 2018], where no regularization is needed due to the low-dimensionality of the data, and (2) [Wainright 2009], where the objective function is simple due to the absence of truncation. In order to deal with both truncation and high-dimensionality at the same time, we develop new techniques that not only generalize the existing ones but we believe are of independent interest.
Spike-and-slab priors are popular Bayesian solutions for high-dimensional linear regression problems. Previous theoretical studies on spike-and-slab methods focus on specific prior formulations and use prior-dependent conditions and analyses, and thus can not be generalized directly. In this paper, we propose a class of generic spike-and-slab priors and develop a unified framework to rigorously assess their theoretical properties. Technically, we provide general conditions under which generic spike-and-slab priors can achieve the nearly-optimal posterior contraction rate and the model selection consistency. Our results include those of Narisetty and He (2014) and Castillo et al. (2015) as special cases.
In this paper, a general algorithm is proposed for rate analysis and code design of linear index coding problems. Specifically a solution for minimum rank matrix completion problem over finite fields representing the linear index coding problem is devised in order to find the optimum transmission rate given vector length and size of the field. The new approach can be applied to both scalar and vector linear index coding.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
