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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf,len=923
+page_content='Statistical whitening of neural populations with gain-modulating interneurons  Lyndon R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Duong * 1 David Lipshutz * 2 David J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Heeger 1 Dmitri B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Chklovskii 2 3 Eero P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Simoncelli 1 2  Abstract  Statistical whitening transformations play a fun-  damental role in many computational systems,  and may also play an important role in biologi-  cal sensory systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Individual neurons appear  to rapidly and reversibly alter their input-output  gains, approximately normalizing the variance of  their responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Populations of neurons appear  to regulate their joint responses, reducing correla-  tions between neural activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' It is natural to see  whitening as the objective that guides these be-  haviors, but the mechanism for such joint changes  is unknown, and direct adjustment of synaptic in-  teractions would seem to be both too slow, and in-  sufficiently reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Motivated by the extensive  neuroscience literature on rapid gain modulation,  we propose a recurrent network architecture in  which joint whitening is achieved through modu-  lation of gains within the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Specifically, we  derive an online statistical whitening algorithm  that regulates the joint second-order statistics of a  multi-dimensional input by adjusting the marginal  variances of an overcomplete set of interneuron  projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' The gains of these interneurons are  adjusted individually, using only local signals,  and feed back onto the primary neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' The net-  work converges to a state in which the responses  of the primary neurons are whitened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We demon-  strate through simulations that the behavior of the  network is robust to poor conditioning or noise  when the gains are sign-constrained, and can be  generalized to achieve a form of local whitening  in convolutional populations, such as those found  throughout the visual or auditory system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Equal contribution 1Center for Neural Science, New York Uni-  versity, New York, NY 2Center for Computational Neuroscience,  Flatiron Institute, New York, NY 3Neuroscience Institute, New  York University School of Medicine, New York, NY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Correspon-  dence to: Lyndon R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Duong <lyndon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='duong@nyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='edu>, David  Lipshutz <dlipshutz@flatironinstitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='org>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Introduction  Statistical whitening transformations, in which multi-  dimensional inputs are decorrelated and normalized to have  unit variance, are common in statistical signal processing  and machine learning systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' For example, they provide a  common step in statistical factorization methods (Hyv¨arinen  & Oja, 2000) and are often used as a preprocessing step  for training deep networks (Krizhevsky, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Empiri-  cal evidence shows that statistical whitening improves un-  supervised feature learning (Coates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' More  recently, self-supervised learning methods have used sta-  tistical whitening or related decorrelation transformations  to prevent representational collapse (Ermolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Zbontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Bardes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Hua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Whitening in neural networks is often performed in the of-  fline setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' However, online methods are useful, especially  when the inputs are from dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  In early sensory systems, which receive inputs from dy-  namic environments, changes in sensory input statistics  induce rapid changes in the input-output gains of single  neurons, allowing cells to normalize their output variance  (Fairhall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Nagel & Doupe, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' This is hypoth-  esized to enable maximal information transmission (Barlow,  1961;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Laughlin, 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Fairhall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' At the popu-  lation level, whitening and related adaptive decorrelation  transformations have been reported in sensory areas such  as the early visual cortex of cats (Benucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2013) and  the olfactory bulb in zebrafish (Friedrich, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Wanner &  Friedrich, 2020) and mice (Giridhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Gschwend  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' However, the mechanisms underlying such  whitening behaviors are unknown, and would seem to re-  quire coordination among all pairs of neurons, as opposed to  the single-neuron case which relies only on gain rescaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Here, motivated by the large neuroscience literature on rapid  gain modulation, we propose a novel recurrent network ar-  chitecture for statistical whitening that exclusively relies on  gain modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' In particular, we introduce a novel objec-  tive for statistical whitening that is expressed solely in terms  of the marginal variances of an overcomplete representation  of the input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We derive a recurrent circuit to optimize  the objective, and show that it corresponds to a network  comprising primary neurons and an auxiliary population of  interneurons with scalar gain modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Importantly, the  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='11955v1  [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='NC]  27 Jan 2023  Statistical whitening of neural populations with gain-modulating interneurons  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Schematic of a recurrent statistical whitening network with 2 primary neurons and 3 interneurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Left: 2D Scatter plot of the  (non-Gaussian) network inputs x = (x1, x2) whose covariance is the ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Center: Primary neurons, whose outputs are y = (y1, y2),  receive external feedforward inputs, x, and recurrent feedback inputs from an auxiliary population of interneurons, − �3  i=1 giziwi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Linear projection vectors {w1, w2, w3} ∈ R2 encode non-negative feedforward synaptic weights connecting the primary neurons to  interneuron i = 1, 2, 3 (symmetric weights are used for feedback connections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' The weights are shown in the left and right panels with  corresponding colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Inset: The ith interneuron (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' here i = 2) receives input zi = w⊤  i y, which is multiplied by its gain gi to produce  output gizi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Its gain, gi, is adjusted s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' ∆gi ∝ z2  i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' The dark arrow indicates that the gain update operates on a slower time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Right: Scatter plots of the whitened network outputs y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Outputs have unit variance along all wi’s, which is equivalent to having identity  covariance matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', Cyy = IN (black circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  network operates online, and its responses converge to the  classical ZCA whitening solution without supervision or  backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' To demonstrate potential applications of  this framework, we show that gain modulation serves as an  implicit gating mechanism, which facilitates fast context-  dependent whitening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Further, we show how non-negative  gain modulation provides a novel approach for dealing with  ill-conditioned or noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Finally, we relax the overcom-  pleteness constraint in our objective and provide a method  for local decorrelation of convolutional populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' A novel objective for ZCA whitening  Consider a neural network with N primary neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' For  each t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , let xt and yt be N-dimensional vectors  whose components respectively denote the inputs and out-  puts of the primary neurons at time t, Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Without loss  of generality we assume the inputs xt are centered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Conventional objective  Statistical whitening aims to linearly transform inputs xt so  that the covariance of the outputs yt is identity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=',  Cyy = ⟨yty⊤  t ⟩t = IN,  (1)  where ⟨·⟩t denotes the expectation operator over t, and IN  denotes the N × N identity matrix (see Appendix A for a  list of notation used in this work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  It is well known that whitening is not unique: any orthog-  onal rotation of a random vector with identity covariance  matrix also has identity covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' There are sev-  eral common choices to resolve this rotational ambiguity,  each with their own advantages (Kessy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Here,  we focus on the popular whitening transformation called  Zero-phase Component Analysis (ZCA) whitening or Ma-  halanobis whitening, which is the whitening transformation  that minimizes the mean-squared error between the inputs  and the whitened outputs (alternatively, the one whose trans-  formation matrix is symmetric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Mathematically, the ZCA-  whitened outputs are the optimal solution to the minimiza-  tion problem  min  {yt}⟨∥xt − yt∥2  2⟩t  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  ⟨yty⊤  t ⟩t = IN,  (2)  where ∥ · ∥2 denotes the Euclidean norm on RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Assuming  the covariance of the inputs Cxx := ⟨xtx⊤  t ⟩t is positive  definite, the unique solution to the optimization problem  in Equation 2 is yt = C−1/2  xx  xt for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , where  C−1/2  xx  is the inverse matrix square root of Cxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Equation 2 provides a starting point for deriving online ZCA  whitening algorithms that can be implemented with recur-  rent neural networks that learn by updating their synaptic  weights (Pehlevan & Chklovskii, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' A novel objective using marginal statistics  We formulate a novel objective for learning the ZCA whiten-  ing transform via gain modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Our innovation exploits  the fact that a random vector has identity covariance matrix  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', Equation 1 holds) if and only if it has unit marginal  g12  W2,1  9222  1  1  W:  92之。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  W2,2  Close-up of  gain-modulated interneuron  Input:(1, 2Statistical whitening of neural populations with gain-modulating interneurons  variance along all possible 1D projections (a form of to-  mography;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' see Related Work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We can derive a tighter  statement, that holds for a finite but overcomplete set of at  least K ≥ KN := N(N + 1)/2 distinct axes (‘overcom-  plete’ simply means that the number of axes exceeds the  dimensionality of the input, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', K > N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Intuitively, this  equivalence holds because an N × N symmetric matrix has  KN degrees of freedom, so the marginal variances along  K ≥ KN distinct axes are sufficient to constrain the N ×N  (symmetric) covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We formalize this equiva-  lence in the following proposition, whose proof is provided  in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Fix K ≥ KN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Suppose w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , wK ∈  RN are unit vectors1 such that  span({w1w⊤  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , wKw⊤  K}) = SN,  (3)  where SN denotes the KN-dimensional vector space of N ×  N symmetric matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Then Equation 1 holds if and only if  the projection of yt onto each unit vector w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , wK has  unit variance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=',  ⟨(w⊤  i yt)2⟩t = 1  for  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  (4)  Assuming Equation 3 holds, we can interpret the set of  vectors {w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , wK} as a frame (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', an overcomplete  basis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Casazza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2013) in RN such that the covariance  of the outputs Cyy can be computed from the variances of  the K-dimensional projection onto the set of frame vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Thus, we can replace the whitening constraint in Equation 2  with the equivalent marginal variance constraint to obtain  the following objective:  min  {yt}⟨∥xt − yt∥2  2⟩t  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Equation 4 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  (5)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' A recurrent neural network with gain  adaptation for ZCA whitening  In this section, we derive an online algorithm for solving  the optimization problem in Equation 5 and map the algo-  rithm onto a recurrent neural network with gain modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  We first introduce Lagrange multipliers to enforce the con-  straints, which transforms the minimization problem into a  minimax problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We then solve the minimax problem by  taking stochastic gradient steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Assume we have an overcomplete frame {w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , wK} in  RN satisfying Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We concatenate the frame vectors  into an N ×K matrix W := [w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , wK].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' In our network,  primary neurons project onto the layer of K interneurons  with the synaptic weights representing matrix W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Then,  the post-synaptic currents in interneurons at time t encode  1The unit-length assumption is without loss of generality and  is imposed here for notational convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  the K-dimensional vector zt := W⊤yt (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We  emphasize that the synaptic weight matrix W will remain  fixed in our whitening algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Enforcing the marginal variance constraints with  scalar gains  We introduce Lagrange multipliers g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , gK ∈ R to en-  force the K constraints in Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We concatenate  the Lagrange multipliers into the K-dimensional vector  g := [g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , gK]⊤ ∈ RK, and formulate the problem as a  saddle point optimization,  max  g  min  {yt}⟨ℓ(xt, yt, g)⟩t,  (6)  where ℓ(x, y, g) := ∥x − y∥2  2 +  K  �  i=1  gi  �  (w⊤  i y)2 − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Here, we have interchanged the order of maximization over  g and minimization over yt, which is justified because  ℓ(xt, yt, g) is convex in yt and linear in g, see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  In our neural network implementation, gi will correspond  to the multiplicative gain associated with the ith interneuron,  so that its output at time t is gizi,t (Figure 1, Inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' From  Equation 6, we see that the gain of the ith interneuron, gi,  enforces the marginal variance of yt along the axis spanned  by wi to be unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Importantly, the gains are not hyper-  parameters, but rather they are optimization variables which  promote statistical whitening of {yt}, preventing the neural  outputs from trivially matching the inputs {xt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Deriving recurrent neural network update rules  To solve Equation 6 in the online setting, we assume there  is a time-scale separation between ‘fast’ neural dynamics  and ‘slow’ gain updates, so that at each time step the neural  dynamics equilibrate before the gains are adjusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' This al-  lows us to perform the inner minimization over {yt} before  the outer maximization over the gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' In biological neural  networks, this is justifiable because a given neuron’s activa-  tions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' action potential firing) operate on a much more  rapid time-scale than its intrinsic input-output gain, which  is driven by slower processes such as changes in calcium  ion concentration gradients (Ferguson & Cardin, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' FAST NEURAL ACTIVITY DYNAMICS  For each time step t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , we minimize the objective  ℓ(xt, yt, g) over yt by recursively running gradient-descent  steps to equilibrium:  yt ← yt − γ  2 ∇yℓ(xt, yt(τ), g)  = yt + γ {xt − W(g ◦ zt) − yt} ,  (7)  where γ > 0 is a small constant, the circle ‘◦’ denotes the  Hadamard (element-wise) product, g ◦ zt is a vector of K  Statistical whitening of neural populations with gain-modulating interneurons  gain-modulated interneuron outputs, and we assume the  primary cell outputs are initialized at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  We see from the right-hand-side of Equation 7 that the ‘fast’  dynamics of the primary neurons are driven by three terms  (inside the curly braces): i) constant feedforward external  input xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' ii) recurrent gain-modulated feedback from in-  terneurons −W(g ◦ zt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' and iii) a leak term −yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Because  the neural activity dynamics are linear, we can analytically  solve for their equilibrium (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' steady-state), ¯yt, by setting  the update in Equation 7 to zero:  ¯yt =  �  IN + W diag (g) W⊤�−1 xt  =  �  IN +  K  �  i=1  giwiw⊤  i  �−1  xt,  (8)  where diag (g) denotes the K × K diagonal matrix whose  (i, i)th entry is gi, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' The equilibrium feed-  forward interneuron inputs are then given by  ¯zt = W⊤¯yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  (9)  The gain-modulated outputs of the K interneurons, g ◦ zt,  are then projected back onto the primary cells via symmetric  weights, −W (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' SLOW GAIN DYNAMICS  After the fast neural activities reach steady-state, the in-  terneuron gains are updated by taking a stochastic gradient-  ascent step with respect to g:  g ← g + η  2∇gℓ(xt, ¯yt, g)  = g + η  �¯z◦2  t − 1  �  ,  (10)  where η  >  0 is the learning rate, the superscript  ‘◦2’ denotes the element-wise squaring operation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=',  ¯z◦2  t  = [¯z2  t,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , ¯z2  t,K]⊤) and 1 = [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , 1]⊤ is the K-  dimensional vector of ones2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Remarkably, the update to the  ith interneuron’s gain gi (Equation 10) depends only on the  online estimate of the variance of its equilibrium input ¯z2  t,i,  and its distance away from the target variance, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Networks  such as these which adapt using only local signals to each  interneuron are suitable candidates for hardware implemen-  tations using low-power neuromorphic chips (Pehlevan &  Chklovskii, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Thus, although statistical whitening in-  herently requires a joint transformation in response to joint  statistics, our recurrent network solution operates solely  using single-neuron gain changes in response to marginal  statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  2Appendix D generalizes the gain update to allowing for  temporal-weighted averaging of the variance over past samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' ONLINE UNSUPERVISED ALGORITHM  By combining Equations 7 – 10, we arrive at our online  recurrent neural network algorithm for statistical whitening  via gain modulation (Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We also provide batched  and offline versions of the algorithm in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Algorithm 1 Online ZCA whitening via gain modulation  1: Input: Centered inputs x1, x2, · · · ∈ RN  2: Initialize: W ∈ RN×K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' g ∈ RK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' η, γ > 0  3: for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' do  4:  yt ← 0  5:  {Run yt and zt dynamics to equilibrium}  6:  while not converged do  7:  zt ← W⊤yt  8:  yt ← yt + γ {xt − W(g ◦ zt) − yt}  9:  end while  10:  g ← g + η  �  z◦2  t − 1  �  {Update gains}  11: end for  There are a few points worth noting about this network:  The weights W remain fixed in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Rather,  the gains g adapt to statistically whiten the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  This allows the whitening to be easily adjusted and  reversed, by simply returning the gains to their default  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  While the objective is effectively in the form of an auto-  encoding loss function involving an ℓ2 reconstruction  term (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' 6), the recurrent network never explicitly  reconstructs its inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Since all recurrent dynamics are linear, it is possible to  bypass the inner loop representing the fast dynamics  of the primary cells (lines 6 – 9 of Algorithm 1), by  directly computing the equilibrium responses of ¯yt,  and ¯z directly (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' 8, 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Numerical experiments and applications  We provide different applications of our recurrent ZCA  whitening network via gain modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' In particular, we  emphasize that gain adaptation is distinct from, while also  complementary to, a synaptic weight learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We therefore  side-step the goal of learning the frame W, and assume it  is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' This allows us to decouple and analyze the gen-  eral properties of our proposed gain modulation framework,  independently from the choice of frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Gain modulation: a new solution to ZCA  whitening  We first demonstrate that our algorithm succeeds in yielding  statistically whitened outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We simulated a network with  interneuron weights, W, as illustrated in Figure 1 (N=2,  Statistical whitening of neural populations with gain-modulating interneurons  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Network from Figure 1 (with corresponding colors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  N=2, K=KN=3, η=2E-3) whitening to two randomly gener-  ated statistical contexts online (10K steps each).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Top: Marginal  variances (log scale) measured by interneurons approach 1 over  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Middle: Dynamics of interneuron gains, which are applied  to zi before feeding back onto the primary cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Dashed lines are  optimal gains (Appendix F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Bottom: Whitening error over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  K=KN=3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Figure 2 shows network adaptation to inputs  from two contexts with randomly generated underlying in-  put covariances Cxx (10K gain update steps each).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' As up-  date steps progress, all marginal variances converge to unity,  as expected from the objective (top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' To achieve ZCA  whitening at equilibrium, then IN +�K  i=1 giwiw⊤  i = C1/2  xx  (Equation 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' When the number of interneurons satisfies  K=KN, the optimal gains to achieve ZCA whitening can  be solved analytically (see Appendix F for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' These  are displayed as dashed lines in the (middle panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We  found that the network successfully adapted to the two ran-  dom statistical contexts, and converged to the optimal set  of gains to achieve whitened yt (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Accordingly,  the whitening error, as measured by the Frobenius norm be-  tween Cyy and IN, approached zero (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Thus,  with each interneuron monitoring their respective marginal  input variances z2  i , and re-scaling their input-output gains to  modulate feedback onto the primary neurons, the network  succeeded in adapting to each context and yielded whitened  outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Rate of convergence depends on frame W  Thus far, we have assumed the frame, W, was fixed and  known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', optimized through pre-training or long time-  scale development).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' This distinguishes our method from  existing ZCA whitening methods, which typically operate  by estimating the eigenvectors of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' By contrast, our  network obviates learning the principal axes of the data  altogether, and instead uses a statistical sampling approach  along a fixed set of measurement axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  If the number of interneurons K=KN, their gains will de-  scend the gradient of the objective (Equation 10), and by  Proposition Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='1, the outputs will become whitened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  We were interested in how effectively the network whitened  randomly sampled inputs with fixed input covariance de-  pending on its initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Figure 3 summarizes an empir-  ical convergence test of 100 networks where N = 2 with  three different kinds of frame W ∈ RN×KN : i) with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Gaussian entries (‘Random’);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' ii) through an optimization  procedure that finds a frame whose columns have mini-  mum mutual coherence and cover the ambient space (‘Opti-  mized’);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' and iii) a frame whose first N columns were the  eigenvectors of the data and the remaining KN −N columns  were random Gaussian entries (‘Spectral’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' For clarity, we  have removed the effects of sampling stochasticity by run-  ning the offline version of our network, which assumes  having direct access to the input covariance (Appendix E);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  the online version was qualitatively similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Convergence depends on qualitative structure of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Networks each had N=2, K=KN=3, η=5E-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Shaded error  regions are standard errors over the 100 repeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  The Spectral frame defines a bound on achievable perfor-  mance, converging much faster than the Random and Op-  timized frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' This is because the interneuron axes were  aligned with the input’s principal axes, and a simple gain  scaling along those directions is the optimal whitening solu-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Interestingly, we found that networks with optimized  weights systematically converged faster than randomly-  initialized frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' These results indicate that the choice  of frame does in fact play an important role in the effective-  ness of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Namely, increased coverage of the  space by the frame vectors facilitates whitening with our  gain re-scaling mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' The random sampling approach  has little hope of scaling to high dimensional inputs, and the  green line in Figure 3 shows that one would benefit from  aligning the frame vectors to the principal axes of the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Implicit gating via gain modulation  Motivated by the findings in Figure 3, we wished to demon-  strate a way in which our adaptive gain modulation net-  work could complement or augment a network in which  context-dependent weights have already been learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We  performed an experiment involving a network with ‘pre-  trained’ W (N=6, K=KN=21) whitening inputs from  100  2  ICy-Inl  2F  10-2  10-6  0  10000  20000  Stepw  Random  Optimized  10-1  Spectral  10-2  10-3  0  200  400  600  800  1000  StepStatistical whitening of neural populations with gain-modulating interneurons  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Gains can act as an implicit gating mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Top:  Whitening error over time with a network (N=6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' KN=21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' η=1E-  3) adapting to 2 alternating statistical contexts A and B, with  different input covariances for 10K steps each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' W was initialized  as a Spectral frame, with the first 2N columns set to be the eigen-  vectors of covariances of contexts A and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Bottom:  Gains can be seen to act as switches for context, gating the spectral  components to optimally whiten each context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  two alternating statistical contexts, A and B, for 10K steps  each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' The frame was constructed such that the first and  second N columns were the eigenvectors of context A and  B’s covariance, respectively, and the remaining K − 2N  columns’ elements were random i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Figure 4  (top panel) shows that the network adaptively whitens the  inputs from each successive context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Surprisingly, upon  closer inspection to the K interneurons’ gains over time  (bottom panel) showed that they approximately served to  ‘select’ the frame vectors corresponding to the eigenvectors  of each respective condition (as indicated by the blue/red in-  tensity on the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Our gain modulation framework thus  serves as an effective means of gating context-dependent  information without an explicit context signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Normalizing ill-conditioned data  When inputs are low-rank, Cxx is ill-conditioned (Fig-  ure 5A), and whitening can amplify directions of small  variance that are due to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' In this section, we show how  our gain-modulating network can be simply modified to han-  dle these types of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' To prevent amplification of inputs  below a certain threshold, we can replace the unit marginal  variance equality constraints with upper bound constraints:  ⟨(w⊤  i yt)2⟩t ≤ 1  for  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  (11)  Our modified network objective then becomes  min  {yt}⟨∥xt − yt∥2  2⟩t  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Equation 11 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  (12)  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Two networks (N=2, K=3, η=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='02) whitening ill-  conditioned inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' A: Outputs without whitening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' 2D scatterplot  of a non-Gaussian density whose underlying signal lies close to  a latent 1D axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' The signal magnitude along that axis is denoted  by the colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' The covariance matrix is depicted as a black ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Gray dashed lines are axes spanned by W (here chosen to be an  equi-angular frame).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' B: ZCA whitening boosts small-amplitude  noise lying along the uninformative direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' C: Modulating gains  according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' 14 rescales the data without amplifying noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' D:  Gains updated with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' 10 (solid) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' 14 (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Intuitively, if the projected variance along a given direction  is already less than or equal to unity, then it will not affect  the overall loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' To enforce the upper bound constraints,  we introduce gains as Lagrange multipliers as before, but  restrict the domain of g to be the non-negative orthant RK  + ,  resulting in non-negative optimal gains:  max  g∈RK  +  min  {yt}⟨ℓ(xt, yt, g)⟩t,  (13)  where ℓ(x, y, g) is defined as in Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' At each time  step t, we optimize Equation 13 by first taking gradient-  descent steps with respect to yt, resulting in the same neu-  ral dynamics (Equation 7) and equilibrium solution (Equa-  tion 8) as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' After the neural activities equilibrate, we  take a projected gradient-ascent step with respect to g:  g ← ⌊g + η(¯z◦2  t − 1)⌋  (14)  where ⌊·⌋ denotes the element-wise half-wave rectification  operation that projects its inputs onto the positive orthant  RK  + , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', ⌊v⌋ := [max(v1, 0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , max(vK, 0)]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  We simulated a network with gains set to either updates  using unconstrained gains (Equation 10), or rectified gains  (Equation 14), and observed that these two models con-  verged to two different solutions (Figure 5B, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' When  10-2  Context A  Context B  Context A  Context B  10-3  10-4  10-5  0  Interneuron index  5  10  15  20  0  500  1000  1500  2000  2500  3000  3500  4000  Step (x10)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='20  GainA  B  2  2  1  1  0  0  1 -  1  2  2  1  T 2  1  0  1  2  2  1  0  1  2  y1  y1  C  D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='75  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='25  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='00  0  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='50  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='75  gi  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='00  Lgi]  2  1  0  1  2  0  100  200  300  y1  StepStatistical whitening of neural populations with gain-modulating interneurons  gi was not constrained to be non-negative, the network  achieved global whitening, as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' By contrast, the gains  constrained to be non-negative converged to different val-  ues altogether, with one of them converging to zero rather  than becoming negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' The whitening error for this net-  work unsurprisingly converged to a non-zero value with the  non-negative gain constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Thus, with a non-negative  constraint, the network failed to fully whiten y, but in doing  so, it did not amplify the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' In Appendix G we show  additional cases that provide further geometric intuition on  differences between ZCA whitening and non-negative gain  constrained ZCA whitening with our network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Gain modulation enables local spatial  decorrelation  The requirement of KN interneurons to ensure a statisti-  cally white output becomes prohibitively costly for high-  dimensional inputs due to the number of interneurons scal-  ing as O(N 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' This led us to ask: how many interneurons  are needed in practice?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' For natural sensory inputs such  as images, it is well known that inter-pixel correlation is  highly structured, decaying as a function of distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Using  a Gaussian random walk, we simulated gaze fixation and  micro-saccadic eye movements, drawing 12×12 patch sam-  ples from a natural image (Figure 6A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Hateren & Schaaf,  1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We did this for different randomly selected regions  of the image (colors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' The content of each region is quite  different, but the inter-pixel correlation within each context  fell rapidly with distance (Figure 6B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  We relaxed the O(N 2) marginal variance constraint to in-  stead target whitening of spatially local neighborhoods of  primary neurons with image patch inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' That is, the frame  W spanned K < KN axes in RN, but was constructed such  that overlapping neighborhoods of 4 × 4 primary neurons  were decorrelated, each by a population of interneurons that  was ‘overcomplete’ with respect to that neighborhood (see  Appendix H for frame construction details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Importantly,  taking into account convolutional structure dramatically re-  duces the interneuron complexity from O(N 2) → O(N)  (Appendix H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' This frame is still overcomplete (K > N),  but because K<KN, we no longer guarantee at equilibrium  that Cyy = IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  After running this local whitening network on the inputs  drawn from the red context, we found that (Figure 6C): i)  inter-pixel correlations drop within the region specified by  the local neighborhood;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' and ii) surprisingly, correlations at  longer-range are dramatically reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Accordingly, the co-  variance eigenspectrum of the locally whitened outputs was  significantly flatter compared to the inputs (Figure 6D left  vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' right columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We also provide a 1D example in Ap-  pendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We remark that this empirical result is not at all  obvious – that whitening individual overlapping neighbor-  hoods of neurons should produce a more globally whitened  output covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Indeed, studying whether and when a  globally whitened solution is possible from whitening of  spatial overlapping neighborhoods is an interesting problem  that is worth pursuing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Related work  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Biologically plausible whitening networks  Biological circuits operate in the online setting and, due  to physical constraints, learn exclusively using local sig-  nals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Therefore, to plausibly model neural computation, a  neural network model must operate in the online setting  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', streaming data) and use local learning rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' There  are a few existing normative models of statistical whiten-  ing and related transformations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' however, these models use  synaptic plasticity mechanisms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', changing W) to adapt  to changing input statistics (Pehlevan & Chklovskii, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Pehlevan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Chapochnikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Lipshutz  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Adaptation of neural population responses to  changes in sensory inputs statistics occurs rapidly, on the  order of seconds (Benucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Wanner & Friedrich,  2020), so it could potentially be accounted for by short-  term synaptic plasticity, which operates on the timescale of  tens of milliseconds to minutes (Zucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2002), but  not by long-term synaptic plasticity, which operates on the  timescale of minutes or longer (Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Here,  we explore the alternative hypothesis that modulation of  neural gains, which operates on the order of tens of mil-  liseconds to minutes (Fairhall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2001), facilitates rapid  adaptation of neural populations to changing input statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Tomography and “sliced” density measurements  Our leveraging of 1D projections to compute the ZCA  whitening transform is reminiscent of approaches taken in  the field of tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Geometrically, our method repre-  sents an ellipsoid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', the N dimensional covariance ma-  trix) using noisy 1D projections of the ellipsoid onto axes  spanned by frame vectors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', estimates of the marginal  variances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' This is a special case of reconstruction problems  that have been studied in geometric tomography (Karl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=',  1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Gardner, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' An important distinction between  tomographic reconstruction and our solution to ZCA whiten-  ing is that we are not using the 1D projections to reconstruct  the multi-dimensional inputs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' instead, we are utilizing the  univariate measurements to transform the ellipsoid into a  new shape (a hyper-sphere, in the case of whitening).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  In optimal transport, “sliced” methods offer a way to mea-  sure otherwise intractable p-Wasserstein distances in high  dimensions (Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2015), thereby enabling its use  in optimization loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Sliced methods compute  Wasserstein distance by repeatedly taking series of 1D pro-  jections of two densities, then computing the expectation  Statistical whitening of neural populations with gain-modulating interneurons  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Local spatial whitening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' A) Large grayscale image from which 12×12 image patch samples are drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Colors represent  random-walk sampling from regions of the image corresponding to contexts with different underlying statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Six samples from  each context are shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' B) Without whitening, mean pairwise output pixel correlations decay rapidly with spatial distance in  each context, suggesting that local whitening may be effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' C) Pairwise output pixel correlation of patches from the red context  before (gray) and after global (black dots) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' convolutional whitening with overlapping 4×4 neighborhoods (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Shaded regions  represent standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' D) Top: Expected correlation matrices of all flattened patches of the red context before whitening, and after  global/local ZCA whitening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Correlation and not covariance matrices are displayed here to facilitate comparison;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' all panels use the same  color scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Bottom: Corresponding covariance eigenspectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  over all 1D Wasserstein distances, for which there exists  an analytic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Notably, the 2-Wasserstein distance  between a 1D zero-mean Gaussian with variance σ2 and a  standard normal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' white) density is  W2  �  N  �  0, σ2�  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' N (0, 1)  �  = ∥σ − 1∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Comparing this with the rule by which we update each in-  terneuron gain, gi ← gi + η((w⊤  i ¯yt)2 − 1) (Equation 10),  reveals striking similarity between our recurrent neural net-  work and methods optimizing using sliced Wasserstein dis-  tances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' However, distinguishing characteristics of our ap-  proach include: 1) minimizing distance between univariate  variances rather than standard deviations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' 2) the directions  along which we compute slices (columns of W) are fixed,  whereas sliced methods typically compute a new set of  random projections at each optimization step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' 3) most im-  portantly, our network operates online, and minimizes sliced  variance distances without backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Discussion  We have derived a novel family of recurrent models for  whitening, which use gain modulation to transform joint  second-order statistics of their inputs based on marginal  variance measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We showed that, given sufficiently  many marginal measurements along unique axes, the net-  work will produce ZCA whitened outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' In particular,  our objective (Equation 5) provides an elegant way to think  about the classical problem of statistical whitening, and  draws connections to old concepts in tomography and trans-  port theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' The framework developed here is flexible, with  several generalizations or extensions that we omitted due  to space limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' For example, by replacing the unity  marginal variance constraint by a set of target variances  differing from 1, the network can be used to transform (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  transport) its input density to one matching the correspond-  ing (non-white) covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Modulating feature gains has proven effective in adapting  pre-trained neural networks to novel inputs with out-of-  training distribution statistics (Ball´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Duong  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Mohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' In fact, adaptive gain  modulation is an old concept in neuroscience which we  believe would be of importance to the broader machine  learning community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' In real neural networks, there exist  several computational processes operating concurrently at  different time-scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Examples include synaptic weights  encoding long-term information, while faster processes like  gain modulation facilitate rapid adaptation to different con-  texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Indeed, the demonstrations in this study were largely  agnostic to the exact structure of the weights W, and in-  stead focused on the computational role of adaptive gain  modulation itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We showed how gains can adaptively  decorrelate a network’s outputs without modifying its pre-  trained weights in an online setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Specifically, we showed  that gain modulation: 1) enables fast switching between pre-  learned context-dependent weight regimes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' 2) can be used  in conjunction with properly-aligned interneuron projec-  tion weights to handle ill-conditioned inputs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' and 3) reduce  A  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='0  lation  Unwhitened  Global whitening  Local whitening  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='5  Correl:  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='0  7  7  0  6  12  Distance (px)  102  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='0 b  Unwhitened  Locally whitened  101↓  1011  1011  igenvalue  Globally whitened  Correlation  10°1  oOT  10°1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='5  10-1 -  10-11  LEF  10-2 1  10-2 1  T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  72  144  1  72  144  1  72  144  1  Eigenvector index  0  6  12  Distance (px)Statistical whitening of neural populations with gain-modulating interneurons  long-range dependencies by modifying local signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Feature whitening and decorrelation has become an im-  portant objective constraint in self-supervised contrastive  learning methods to help prevent representational collapse  (Bardes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Zbontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Ermolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=',  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We believe that the networks developed in this study,  motivated by extensive neuroscience research on rapid gain  modulation, provide an effective whitening solution for  these methods – particularly in regimes which prioritize  streaming data, and networks designed for low-power con-  sumption hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  References  Ball´e, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
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+page_content='  Bardes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
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+page_content=', and LeCun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' VICReg: Variance-  invariance-covariance regularization for self-supervised  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
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+page_content=' ISSN 1053-5888, 1558-  0792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
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+page_content=' Whitening of odor  representations by the wiring diagram of the olfactory  bulb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Nature neuroscience, 23(3):433–442, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
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+page_content=' Bar-  low twins: Self-supervised learning via redundancy reduc-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' In International Conference on Machine Learning,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
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+page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Zucker, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', Regehr, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Short-term synaptic  plasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Annual Review of Physiology, 64(1):355–405,  2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Statistical whitening of neural populations with gain-modulating interneurons  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Notation  For N ≥ 2, let KN := N(N + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Let RN denote N-dimensional Euclidean space equipped with the Euclidean norm,  denoted ∥·∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Let RN  + denote the non-negative orthant in RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Given K ≥ 2, let RN×K denote the set of N ×K real-valued  matrices and SN denote the set of N × N symmetric matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Matrices are denoted using bold uppercase letters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', M) and vectors are denoted using bold lowercase letters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Given a matrix M, Mij denotes the entry of M located at the ith row and jth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Let 1 = [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , 1]⊤ denote the  N-dimensional vector of ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Let IN denote the N × N identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Given vectors v, w ∈ RN, define their Hadamard product by v ◦ w := (v1w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , vNwN) ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Define v◦2 :=  (v2  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , v2  N) ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Define diag(v) to be the N × N diagonal matrix whose (i, i)th entry is equal to vi, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Let ⟨·⟩t denote expectation over t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  The diag (·) operator, similar to numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='diag() or MATLAB’s diag(), can either: 1) map a vector in RK to the  diagonal of a K × K zeros matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' or 2) map the diagonal entries of a K × K matrix to a vector in RK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' The specific  operation being used should be clear by context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='1  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Suppose Equation 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Then, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , K,  ⟨(w⊤  i yt)2⟩t = ⟨w⊤  i yty⊤  t wi⟩t = w⊤  i wi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Therefore, Equation 4 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Now suppose Equation 4 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Let v ∈ RN be an arbitrary unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Then vv⊤ ∈ SN and by Equation 3, there exist  g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , gK ∈ R such that  vv⊤ = g1w1w⊤  1 + · · · + gKwKw⊤  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  (15)  We have  v⊤⟨yty⊤  t ⟩tv = Tr(vv⊤⟨yty⊤  t ⟩t) =  K  �  i=1  gi Tr(wiw⊤  i ⟨yty⊤  t ⟩t) =  K  �  i=1  gi Tr(wiw⊤  i ) = Tr(vv⊤) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  (16)  The first equality is a property of the trace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' The second and fourth equalities follows from Equation 15 and the  linearity of the trace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' The third equality follows from Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' The final equality holds because v is a unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Since Equation 16 holds for every unit vector v ∈ RN, Equation 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Saddle point property  We recall the following minmax property for a function that satisfies the saddle point property (Boyd & Vandenberghe, 2004,  section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Let V ⊆ Rn, W ⊆ Rm and f : V × W → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Suppose f satisfies the saddle point property;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' that is, there  exists (a∗, b∗) ∈ V × W such that  f(a∗, b) ≤ f(a∗, b∗) ≤ f(a, b∗),  for all (a, b) ∈ V × W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Then  min  a∈V max  b∈W f(a, b) = max  b∈W min  a∈V f(a, b) = f(a∗, b∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Weighted average update rule for gi  The update for g in Equation 10 can be generalized to allow for a weighted average over past samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' In particular, the  general update is given by  g ← g + η  �  1  Z  t  �  s=1  γt−sz◦2  s − 1  �  ,  Statistical whitening of neural populations with gain-modulating interneurons  where γ ∈ [0, 1] determines the decay rate and Z := 1 + γ + · · · + γt−1 is a normalizing factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Batched and offline algorithms for whitening with RNNs via gain modulation  In addition to the fully-online algorithm provided in the main text (Algorithm 1), we also provide two variants below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' In  many applications, streaming inputs arrive in batches rather than one at a time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' video streaming frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Similarly  for conventional offline stochastic gradient descent training, data is sampled in batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Algorithm 2 would be one way to  accomplish this in our framework, where the main difference between the fully online version is taking the mean across  samples in the batch to yield average gain update ∆g term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Furthermore, in the fully offline setting when the covariance of  the inputs, Cxx is known, Algorithm 3 presents a way to whiten the covariance directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Algorithm 2 Batched ZCA whitening  1: Input: Data matrix X ∈ RN×T (assumed centered)  2: Initialize: W ∈ RN×K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' g ∈ RK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' batch size B  3: while not converged do  4:  XB ← sample batch(X, B){N × B}  5:  Yb ← [IN + W diag (g) W⊤]−1XB  6:  Zb ← W⊤Yb  7:  ∆g ← Z◦2  B − 1 {Subtract 1 from all entries}  8:  g ← g + η mean(∆g, axis=1)  9: end while  Algorithm 3 Offline ZCA whitening  1: Input: Input covariance Cxx  2: Initialize: W ∈ RN×K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' g ∈ RK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' η  3: while not converged do  4:  M ← [IN + W diag (g) W⊤]−1  5:  Cyy ← MCxxM⊤  6:  ∆g ← diag  �  W⊤CyyW  �  − 1  7:  g ← g + η∆g  8: end while  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Frame factorizations of symmetric matrices  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Analytic solution for the optimal gains  Recall that the optimal solution of the ZCA objective in Equation 5 is given by yt = C−1/2  xx  xt for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' In our  neural circuit with interneurons and gain control, the outputs of the primary neurons at equilibrium is (given in Equation 8,  but repeated here for clarity)  ¯yt =  �  IN + W diag (g) W⊤�−1 xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Therefore, the circuit performs ZCA whitening when the gains g satisfy the relation  IN + W diag (g) W⊤ = C1/2  xx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  (17)  When K is exactly N(N + 1)/2, we can explicitly solve for the optimal gains ¯g (derived in the next subsection):  ¯g =  ��  W⊤W  �◦2�−1 �  w⊤  1 C1/2  xx w1 − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , w⊤  NC1/2  xx wN − 1  �⊤  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  (18)  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Deriving optimal gains  We find it useful to first demonstrate that any matrix C ∈ SN, where SN is the space of symmetric N × N matrices, can be  factorized as  W diag (g) W⊤ = C  (19)  where W ∈ RN×K is some fixed, arbitrary, frame with K ≥ N(N+1)  2  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' a representation that is O(N 2) overcomplete),  and g ∈ RK is a variable vector encoding information about C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' We multiply both sides of Equation 19 from the left and  right by W⊤ and W, respectively, then take the diagonal3 of the resultant matrices,  diag  �  W⊤W diag (g) W⊤W  �  = diag  �  W⊤CW  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  (20)  3Similar to commonly-used matrix libraries, the diag (·) operator here is overloaded and can map a vector to a matrix or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  See Appendix A for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Statistical whitening of neural populations with gain-modulating interneurons  Finally, employing a simple matrix identity involving the diag (·) operator yields  (W⊤W)◦2g = diag  �  W⊤CW  �  ,  (21)  =⇒ g =  �  (W⊤W)◦2�−1 diag  �  WT CW  �  ,  (22)  where (·)◦2 denotes element-wise squaring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Thus, any N × N symmetric matrix, can be encoded as a vector, g, with respect  to an arbitrary fixed frame, W, by solving a standard linear system of K equations of the form Ag = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Importantly, when  K = N(N + 1)/2, and the columns of W are not collinear, then the matrix on the LHS, (W⊤W)◦2 ∈ SK  ++, is invertible,  and the vector g is unique (Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Without loss of generality, assume that the columns of W are unit-norm (otherwise, we can always normalize them by  absorbing their lengths into the elements of g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Furthermore, assume without loss of generality that C ∈ SN  ++, the set of all  symmetric positive definite matrices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' covariance, precision, PSD square roots, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' When C is a covariance matrix,  then diag  �  W⊤CW  �  can be interpreted as a vector of projected variances of C along each axis spanned by W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Therefore,  Equation 21 states that the vector g is linearly related to the vector of projected variances via the element-wise squared  frame Gramian, (W���W)◦2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Adaptation with inequality constraint  In general, the modified objective with rectified gains (Equation 14) does not statistically whiten the inputs x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' ,  but rather adapts the non-negative gains g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , gK to ensure that the variances of the outputs y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' in the directions  spanned by the frame vectors {w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , wK} are bounded above by unity (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' This one-sided normalization  carries interesting implications for how and when the circuit statistically whitens its outputs, which can be compared with  experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' For instance, the circuit performs ZCA whitening if and only if there are non-negative gains such  that Equation 17 holds (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', the top right example in Figure 7), which corresponds to cases such that the matrix C1/2  xx is  an element of the following cone (with its vertex translated by IN):  �  IN +  K  �  i=1  giwiw⊤  i : g ∈ RK  +  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  On the other hand, if the variance of an input projection is less than unity — i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', w⊤  i Cxxwi ≤ 1 for some i — then the  corresponding gain gi remains zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' When this is true for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , K, the gains all remain zero and the circuit output  is equal to its input (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=', the bottom middle example of Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Geometric intuition of whitening with/without inequality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Whitening efficacy using non-negative gains depends on W  and Cxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' For N = 2 and K = 3, examples of covariance matrices Cyy (red ellipses) corresponding to optimal solutions y of objective  12, for varying input covariance matrices Cxx (black ellipses) and frames W (spanning axes denoted by gray lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Unit circles, which  correspond to the identity matrix target covariance, are shown with dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Each row corresponds to a different frame W and each  column corresponds to a different input covariance Cxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Cxx,1  Cxx,2  Cxx,3  IM  W2Statistical whitening of neural populations with gain-modulating interneurons  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Whitening spatially local neighborhoods  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Spatially local whitening in 1D  For an N-dimensional input, we consider a network that whitens spatially local neighborhoods of size M < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' To this end,  we can construct N filters of the form  wi = ei,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , N  and M(N − M+1  2  ) filters of the form  w = ei + ej  √  2  ,  i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' , N,  1 ≤ |i − j| ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  The total number of filters is (M + 1)(N − M  2 ), so for fixed M the number of filters scales linearly in N rather than  quadratically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  We simulated a network comprising N = 10 primary neurons, and a convolutional weight matrix connecting each interneuron  to spatial neighborhoods of three primary neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Given input data with covariance Cxx illustrated in Figure 8A (left  panel), this modified network succeeded to statistically whiten local neighborhoods of size of primary 3 neurons (right  panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Notably, the eigenspectrum (Figure 8B) after local whitening is much closer to being equalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Furthermore, while  the global whitening solution produced a flat spectrum as expected, the local whitening network did not amplify the axis  with very low-magnitude eigenvalues (Figure 8B right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Statistically adapting local neighborhoods of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' A) ˆCxx denotes correlation matrix, which are shown here for display  purposes only, to facilitate comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Network with 10-dimensional input correlation (left) 10-dimensional output correlation matrix  after global whitening (middle);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' and output correlation matrix after statistically whitening local neighborhoods of size 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' The output  correlation matrix of the locally adapted circuit has block-identity structure along the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' B) Corresponding eigenspectra of  covariance matrices of unwhitened (left), global whitened (middle), and locally whitened (right) network outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' The black dashed line  denotes unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Filter bank construction in 2D  Here, we describe one way of constructing a set of convolutional weights for overlapping spatial neighborhoods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' image  patches) of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' Given an n × m input and overlapping neighborhoods of size h × w to be statistically whitened, the  samples are therefore matrices X ∈ Rn×m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content=' In this case, filters w ∈ R1×n×m can be indexed by pairs of pixels that are in  the same patch:  ((i, j), (k, ℓ)),  1 ≤ i ≤ n,  1 ≤ j ≤ m,  0 ≤ |i − k| ≤ h,  0 ≤ |j − ℓ| ≤ w  A  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='0  Cyy globally white  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='0  yy locally white  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
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+page_content='0  B  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
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+page_content='0  30  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
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+page_content='5  15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
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+page_content='0  1  5  10  1  5  10  1  5  10  EigenvectorStatistical whitening of neural populations with gain-modulating interneurons  We can then construct the filters as,  w(i,j),(k,ℓ)(X) =  �  xi,j  if (i, j) = (k, ℓ),  xi,j+xk,ℓ  √  2  if (i, j) ̸= (k, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}
+page_content='  In this case there are  nm + wh  �  (n − w)(m − h) + (n − w)(h + 1)  2  + (m − h)(w + 1)  2  + (h + 1)(w + 1)  2  �  such filters, so the number of filters required scales linearly with nm rather than quadratically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf'}