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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Maximally informative dimensions ： ウィキペディア英語版 Maximally informative dimensions Maximally informative dimensions is a dimensionality reduction technique used in the statistical analyses of neural responses. Specifically, it is a way of projecting a stimulus onto a low-dimensional subspace so that as much information as possible about the stimulus is preserved in the neural response. It is motivated by the fact that natural stimuli are typically confined by their statistics to a lower-dimensional space than that spanned by white noise.〔D.J. Field. "Relations between the statistics of natural images and the response properties of cortical cells." J. Opt. Soc. am. A 4:2479-2394, 1987.〕 Within this subspace, however, stimulus-response functions may be either linear or nonlinear. The idea was originally developed by Tatyana Sharpee, Nicole Rust, and William Bialek in 2003.〔Sharpee, Tatyana, Nicole C. Rust, and William Bialek. "Maximally informative dimensions: analyzing neural responses to natural signals." Advances in Neural Information Processing Systems (2003): 277-284.〕 ==Mathematical formulation== Neural stimulus-response functions are typically given as the probability of a neuron generating an action potential, or spike, in response to a stimulus $\backslash mathbf$. The goal of maximally informative dimensions is to find a small relevant subspace of the much larger stimulus space that accurately captures the salient features of $\backslash mathbf$. Let $D$ denote the dimensionality of the entire stimulus space and $K$ denote the dimensionality of the relevant subspace, such that $K\; \backslash ll\; D$. We let $\backslash$ denote the basis of the relevant subspace, and $\backslash mathbf^K$ the projection of $\backslash mathbf$ onto $\backslash$. Using Bayes' theorem we can write out the probability of a spike given a stimulus: : $P(spike|\backslash mathbf^K)\; =\; P(spike)f(\backslash mathbf^K)$ where : $f(\backslash mathbf^K)\; =\; \backslash frac$ is some nonlinear function of the projected stimulus. In order to choose the optimal $\backslash$, we compare the prior stimulus distribution $P(\backslash mathbf)$ with the spike-triggered stimulus distribution $P(\backslash mathbf|spike)$ using the Shannon information. The average information (averaged across all presented stimuli) per spike is given by : $I\_\; =\; \backslash sum\_|spike)\; log\_2\; ()$.〔N. Brenner, S. P. Strong, R. Koberle, W. Bialek, and R. R. de Ruyter van Steveninck. "Synergy in a neural code. Neural Comp., 12:1531-1552, 2000.〕 Now consider a $K\; =\; 1$ dimensional subspace defined by a single direction $\backslash mathbf$. The average information conveyed by a single spike about the projection $x\; =\; \backslash mathbf\; \backslash cdot\; \backslash mathbf$ is : $I(\backslash mathbf)\; =\; \backslash int\; dx\; P\_\}(x|spike)/P\_\}(x|spike)/P\_\}(x|spike)\; =\; \backslash langle\; \backslash delta(x\; -\; \backslash mathbf\; \backslash cdot\; \backslash mathbf)\; |spike\; \backslash rangle\_\}(x)\; =\; \backslash langle\; \backslash delta(x\; -\; \backslash mathbf\; \backslash cdot\; \backslash mathbf)\backslash rangle\_$. Under this formulation, the relevant subspace of dimension $K\; =\; 1$ would be defined by the direction $\backslash mathbf$ that maximizes the average information $I(\backslash mathbf)$. This procedure can readily be extended to a relevant subspace of dimension $K\; >\; 1$ by defining : $P\_(\backslash mathbf|spike)\; =\; \backslash langle\; \backslash prod\_^K\; \backslash delta(x\_i\; -\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\_i)\; |spike\; \backslash rangle\_^K\}(\backslash mathbf)\; =\; \backslash langle\; \backslash prod\_^K\; \backslash delta(x\_i\; -\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\_i)\; \backslash rangle\_^K\})$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Maximally informative dimensions」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.