اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدODE-Inspired Analysis for the Biological Version of Ojas Rule in Solving Streaming PCA
87
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Ojas rule [Oja, Journal of mathematical biology 1982] is a well-known biologically-plausible algorithm using a Hebbian-type synaptic update rule to solve streaming principal component analysis (PCA). Computational neuroscientists have known that this biological version of Ojas rule converges to the top eigenvector of the covariance matrix of the input in the limit. However, prior to this work, it was open to prove any convergence rate guarantee. In this work, we give the first convergence rate analysis for the biological version of Ojas rule in solving streaming PCA. Moreover, our convergence rate matches the information theoretical lower bound up to logarithmic factors and outperforms the state-of-the-art upper bound for streaming PCA. Furthermore, we develop a novel framework inspired by ordinary differential equations (ODE) to analyze general stochastic dynamics. The framework abandons the traditional step-by-step analysis and instead analyzes a stochastic dynamic in one-shot by giving a closed-form solution to the entire dynamic. The one-shot framework allows us to apply stopping time and martingale techniques to have a flexible and precise control on the dynamic. We believe that this general framework is powerful and should lead to effective yet simple analysis for a large class of problems with stochastic dynamics.
قيم البحث
اقرأ أيضاً
Replay is the reactivation of one or more neural patterns, which are similar to the activation patterns experienced during past waking experiences. Replay was first observed in biological neural networks during sleep, and it is now thought to play a
critical role in memory formation, retrieval, and consolidation. Replay-like mechanisms have been incorporated into deep artificial neural networks that learn over time to avoid catastrophic forgetting of previous knowledge. Replay algorithms have been successfully used in a wide range of deep learning methods within supervised, unsupervised, and reinforcement learning paradigms. In this paper, we provide the first comprehensive comparison between replay in the mammalian brain and replay in artificial neural networks. We identify multiple aspects of biological replay that are missing in deep learning systems and hypothesize how they could be utilized to improve artificial neural networks.
In this paper we propose a new algorithm for streaming principal component analysis. With limited memory, small devices cannot store all the samples in the high-dimensional regime. Streaming principal component analysis aims to find the $k$-dimension
al subspace which can explain the most variation of the $d$-dimensional data points that come into memory sequentially. In order to deal with large $d$ and large $N$ (number of samples), most streaming PCA algorithms update the current model using only the incoming sample and then dump the information right away to save memory. However the information contained in previously streamed data could be useful. Motivated by this idea, we develop a new streaming PCA algorithm called History PCA that achieves this goal. By using $O(Bd)$ memory with $Bapprox 10$ being the block size, our algorithm converges much faster than existing streaming PCA algorithms. By changing the number of inner iterations, the memory usage can be further reduced to $O(d)$ while maintaining a comparable convergence speed. We provide theoretical guarantees for the convergence of our algorithm along with the rate of convergence. We also demonstrate on synthetic and real world data sets that our algorithm compares favorably with other state-of-the-art streaming PCA methods in terms of the convergence speed and performance.
We consider the following multi-component sparse PCA problem: given a set of data points, we seek to extract a small number of sparse components with disjoint supports that jointly capture the maximum possible variance. These components can be comput
ed one by one, repeatedly solving the single-component problem and deflating the input data matrix, but as we show this greedy procedure is suboptimal. We present a novel algorithm for sparse PCA that jointly optimizes multiple disjoint components. The extracted features capture variance that lies within a multiplicative factor arbitrarily close to 1 from the optimal. Our algorithm is combinatorial and computes the desired components by solving multiple instances of the bipartite maximum weight matching problem. Its complexity grows as a low order polynomial in the ambient dimension of the input data matrix, but exponentially in its rank. However, it can be effectively applied on a low-dimensional sketch of the data; this allows us to obtain polynomial-time approximation guarantees via spectral bounds. We evaluate our algorithm on real data-sets and empirically demonstrate that in many cases it outperforms existing, deflation-based approaches.
Natural images follow statistics inherited by the structure of our physical (visual) environment. In particular, a prominent facet of this structure is that images can be described by a relatively sparse number of features. We designed a sparse codin
g algorithm biologically-inspired by the architecture of the primary visual cortex. We show here that coefficients of this representation exhibit a heavy-tailed distribution. For each image, the parameters of this distribution characterize sparseness and vary from image to image. To investigate the role of this sparseness, we designed a new class of random textured stimuli with a controlled sparseness value inspired by our measurements on natural images. Then, we provide with a method to synthesize random textures images with a given statistics for sparseness that matches that of some given class of natural images and provide perspectives for their use in neurophysiology.
A popular theory of perceptual processing holds that the brain learns both a generative model of the world and a paired recognition model using variational Bayesian inference. Most hypotheses of how the brain might learn these models assume that neur
ons in a population are conditionally independent given their common inputs. This simplification is likely not compatible with the type of local recurrence observed in the brain. Seeking an alternative that is compatible with complex inter-dependencies yet consistent with known biology, we argue here that the cortex may learn with an adversarial algorithm. Many observable symptoms of this approach would resemble known neural phenomena, including wake/sleep cycles and oscillations that vary in magnitude with surprise, and we describe how further predictions could be tested. We illustrate the idea on recurrent neural networks trained to model image and video datasets. This framework for learning brings variational inference closer to neuroscience and yields multiple testable hypotheses.
الأسئلة المقترحة
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
