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[EQUATION] and [EQUATION] We can then solve for [MATH] and [MATH] and obtain the boundary velocity: [EQUATION] and the boundary stresses: |
[EQUATION] where [MATH] [MATH] and [MATH] are the boundary speed and boundary stress from the homogeneous equation of [MATH] , which are constants. |
To calculate [MATH] , we have to determine [MATH] and [MATH] . Eq. S12 gives one condition [MATH] and an additional condition from the stresses in Eq. S13 is needed. Unfortunately, there is no simple relation between the four equalities in Eq. S13 since the stress continuity equation cannot be defined at the boundary a... |
[EQUATION] Note that when [MATH] , corresponding to [MATH] , this result gives the same scaling as for the simple rectangular shape. |
With [MATH] , corresponding to a highly viscous cytoskeleton, we have [EQUATION] which clearly shows that the cell speed is scaling inversely with the average height [MATH] |
VII Test of model predictions The above analysis indicates that the ratio of the height of stress band [MATH] and the cell height [MATH] determines the cell speed. Thus, cells with equal ratio should have similar speeds. To test this explicitly, we simulated cells in chambers with heights varying between [MATH] and [MA... |
In addition, our derived expression predicts that if [MATH] , corresponding to an active stress region that spans the entire height of the cell, the cell speed should be independent of the chamber height. To verify this, we performed simulations of confined cells with the active stress at the entire cell front. To this... |
VIII Oscillatory protrusions Results in the main text are for cells with constant active stress, resulting in constant cell shapes. Such constant shapes are applicable to fish keratocytes, fast moving cells that maintain their morphology Keren et al. 2008 Other cell types, however, including neutrophils and Dictyosteli... |
[MATH] is the period of the oscillation cycle which can be varied. Results from additional simulations show that the cell speed gets larger as the substrate adhesion is increased (Fig. S12 a). This dependence on adhesion was found to be largely independent of the period and is similar to the one found for model cells w... |
IX Experiments IX.1 Cell culture and preparation Wild type Dictyostelium discoideum (AX4) cells were transformed with a construct in which the regulatory region of actin 15 drives genes encoding a fusion of GFP to LimE ( [MATH] coil LimE-GFP) and a gene encoding a fusion of RFP to Coronin (LimE GFP/corA RFP) Fuller et ... |
IX.2 Microfluidic device The design of microfluidic device used in the study is similar to the design of the devices that were previously used to study gradient sensing in yeast Paliwal et al. 2007 and chemotaxis in Dictyostelium Skoge et al. 2010 2014 . The microfluidic device (Fig. S9 ) consists of a lithographically... |
The lengths [MATH] of the gradient chambers in the upstream, middle, and downstream clusters are 360, 220, or 120 [MATH] m, respectively. There is practically no flow through the gradient chamber because of near zero pressure gradient along them, and the diffusion of cAMP from the flow-through channel perfused with the... |
IX.3 Substrate preparation In our experiments, the microfluidic chips were sealed against cover glass substrates with two different types of coating: [MATH] 10 [MATH] m thick layer of PDMS of the same type as the material of the chip and [MATH] [MATH] m thick layer of 30% polyethylene glycol (PEG) gel. In the former ca... |
IX.4 Data acquisition and image analysis Differential interference contrast (DIC) images were taken of all gradient chambers on a spinning-disk confocal Zeiss Axio Observer inverted microscope using a 10x objective and a Roper Cascade QuantEM 512SC camera. DIC images were captured every 15 s for 30 min and were used to... |
The centroids of all cells were tracked across the gradient chambers from 10X image sequences using Slidebook 6 (Intelligent Imaging Innovations) software. Cells that moved more than 5 frames without encountering another cell were chosen for data analysis. 50 to 100 cell tracks were analyzed in each experiment. Velocit... |
Cells outlines near the top (PDMS chip) and the bottom (substrate, PDMS or PEG) of the gradient chambers were obtained from confocal fluorescence images at 63X magnification with a custom-made Matlab code, as follows. After removing the average background intensity value, images were binarized using a threshold that wa... |
IX.5 Statistics and reproducibility Each experiment was carried out four or five times on different days and the data were averaged for N=200-300 cells. Cell speed was found to be approximately normally distributed and p-values were computed with the unpaired t-test. For the area size ratio, the data distribution was n... |
# Source: arxiv 1806.01316 # Title: Universal Statistics of Fisher Information in Deep Neural Networks: Mean Field Approach # Sections: all # Downloaded: 2026-03-03T05:15:49.504977+00:00 |
Universal Statistics of Fisher Information in Deep Neural Networks: Mean Field Approach Ryo Karakida Shotaro Akaho Shun-ichi Amari |
AIST, Japan AIST, Japan RIKEN CBS, Japan Abstract The Fisher information matrix (FIM) is a fundamental quantity to represent the characteristics of a stochastic model, including deep neural networks (DNNs). The present study reveals novel statistics of FIM that are universal among a wide class of DNNs. To this end, we ... |
Introduction Deep learning has succeeded in making hierarchical neural networks perform excellently in various practical applications |
. To proceed further, it would be beneficial to give more theoretical elucidation as to why and how deep neural networks (DNNs) work well in practice. In particular, it would be useful to not only clarify the individual models and phenomena but also explore various unified theoretical frameworks that could be applied t... |
. For instance, Poole et al. proposed a useful indicator to explain the expressivity of DNNs. Regarding the trainability of DNNs, Schoenholz et al. extended this theory to backpropagation and found that the vanishing and explosive gradients obey a universal law. These studies are powerful in the sense that they do not ... |
Unfortunately, such universal frameworks have not yet been established in many other topics. One is the geometric structure of the parameter space. For instance, the loss landscape without spurious local minima is important for easier optimization and theoretically guaranteed in single-layer models |
, shallow piecewise linear ones , and extremely wide deep networks with the number of training samples smaller than the width Flat global minima have been reported to be related to generalization ability through empirical experiments showing that networks with such minima give better generalization performance |
. However, theoretical analysis of the flat landscape has been limited in shallow rectified linear unit (ReLU) networks Thus, a residual subject of interest is to theoretically reveal the geometric structure of the parameter space truly common among various deep networks. |
To establish the foundation of the universal perspective of the parameter space, this study analytically investigates the Fisher information matrix (FIM). As is overviewed in Section 2.1 , the FIM plays an essential role in the geometry of the parameter space and is a fundamental quantity in both statistics and machine... |
1.1 Main results This study analyzes the FIM of deep networks with random weights and biases, which are widely used settings to analyze the phenomena of DNNs |
First, we analytically obtain novel statistics of the FIM, namely, the mean (Theorem ), variance (Theorem ), and maximum of eigenvalues (Theorem ). These are universal among a wide class of shallow and deep networks with various activation functions. These quantities can be obtained from simple iterative computations o... |
Furthermore, to confirm the potential usage of the derived statistics, we show some exercises. One is on the Fisher-Rao norm (Theorem ). This norm was originally proposed to connect the flatness of a parameter space to the capacity measure of generalization ability. We evaluate the Fisher-Rao norm by using an indicator... |
1.2 Related works Despite its importance in statistics and machine learning, study on the FIM for neural networks has been limited so far. This is because layer-by-layer nonlinear maps and huge parameter dimensions make it difficult to take analysis any further. Degeneracy of the eigenvalues of the FIM has been found i... |
To understand the loss landscape, Pennington and Bahri has utilized random matrix theory and obtained the spectrum of FIM and Hessian under several assumptions, although the analysis is limited to special types of shallow networks. In contrast, this paper is the first attempt to apply the mean field approach, which ove... |
LeCun et al. investigated the Hessian of the loss, which coincides with the FIM at zero training error, and empirically reported that very large eigenvalues exist, i.e., ”big killers”, which affects the optimization (discussed in Section 4.2 ). The eigenvalue distribution peaks around zero while its tail is very long; ... |
, but its theoretical evidence and evaluation have remained unsolved as far as we know. Therefore, our theory provides novel theoretical evidence that this skewed eigenvalue distribution and its huge maximum appear universally in DNNs. |
The theoretical tool we use here is known as the mean field theory of deep networks as briefly overviewed in Section 2.4 This method has been successful in analyzing neural networks with random weights under a large width limit and in explaining the performance of the models. In particular, it quantitatively coincides ... |
. This analysis has been extended from fully connected deep networks to residual and convolutional networks . The evaluation of the FIM in this study is also expected to be extended to such cases. |
Preliminaries 2.1 Fisher information matrix (FIM) We focus on the Fisher information matrix (FIM) of neural network models, which previous works have developed and is commonly used |
. It is defined by [EQUATION] where the statistical model is given by [MATH] . The output model is given by [MATH] , where [MATH] is the network output parameterized by [MATH] and [MATH] is the Euclidean norm. The [MATH] is an input distribution. The expectation [MATH] is taken over the input-output pairs [MATH] of the... |
[MATH] are available, the expectation can be replaced by the empirical mean. This is known as the empirical FIM and often appears in practice |
[EQUATION] This study investigates the above empirical FIM for arbitrary [MATH] It converges to the expected FIM as [MATH] . Although the form of the FIM changes a bit in other statistical models (i.g., softmax outputs), these differences are basically limited to the multiplication of activations in the output layer |
. Our framework can be straightforwardly applied to such cases. The FIM determines the asymptotic accuracy of the estimated parameters, as is known from a fundamental theorem of statistics, namely, the Cramér-Rao bound |
Below, we summarize a more intuitive understanding of the FIM from geometric views. Information geometric view. Let us define an infinitesimal squared distance [MATH] , which represents the Kullback-Leibler divergence between the statistical model [MATH] and [MATH] against a perturbation [MATH] . It is given by |
[EQUATION] It means that the parameter space of a statistical model forms a Riemannian manifold and the FIM works as its Riemannian metric, as is known in information geometry |
. This quadratic form is equivalent to the robustness of a deep network: [MATH] Insights from information geometry have led to the development of natural gradient algorithms |
and, recently, a capacity measure based on the Fisher-Rao norm Loss landscape view. The empirical FIM ( ) determines the local landscape of the loss function around the global minimum. Suppose we have a squared loss function [MATH] . The FIM is related to the Hessian of the loss function, [MATH] , in the following way: |
[EQUATION] The Hessian coincides with the FIM when the parameter converges to the global minimum by learning, that is, the true parameter [MATH] from which the teacher signal [MATH] is [MATH] or, more generally, with noise (i.e., [MATH] , where [MATH] denotes zero-mean Gaussian noise) |
In the literature on deep learning, its eigenvectors whose eigenvalues are close to zero locally compose flat minima, which leads to better generalization empirically |
. Modifying the loss function with the FIM has also succeeded in overcoming the catastrophic forgetting Note that the information geometric view tells us more than the loss landscape. While the Hessian ( ) assumes the special teacher signal, the FIM works as the Riemannian metric to arbitrary teacher signals. |
2.2 Network architecture This study investigates a fully connected feedforward neural network. The network consists of one input layer with [MATH] units, [MATH] hidden layers ( [MATH] ) with [MATH] units per hidden layer [MATH] , and one output layer with [MATH] units: |
[EQUATION] This study focuses on the case of linear outputs, that is, [MATH] We assume that the activation function [MATH] and its derivative [MATH] are square-integrable functions on a Gaussian measure. A wide class of activation functions, including the sigmoid-like and (leaky-) ReLU functions, satisfy these conditio... |
The FIM ( ) of a deep network is computed by the chain rule in a manner similar to that of the backpropagation algorithm: [EQUATION] |
where [MATH] for ( [MATH] ). To avoid the complicated notation, we omit the index of the output unit, i.e., [MATH] , in the following. |
2.3 Random connectivity The parameter set [MATH] is an ensemble [EQUATION] and then fixed, where [MATH] denotes a Gaussian distribution with zero mean and variance [MATH] , and we set [MATH] and [MATH] To avoid complicated notation, we set them uniformly as [MATH] and [MATH] , but they can easily be generalized. It is ... |
We also assume that the input samples [MATH] are generated in an i.i.d. manner from a standard Gaussian distribution: [MATH] We focus here on the Gaussian case for simplicity, although we can easily generalize it to other distributions with finite variances. |
Let us remark that the above random connectivity is a common setting widely supposed in theories. Analyzing such a network can be regarded as the typical evaluation |
It is also equal to analyzing the network randomly initialized The random connectivity is often assumed in the analysis of optimization as a true parameter of the networks, that is, the global minimum of the parameters |
2.4 Mean-field approach On neural networks with random connectivity, taking a large width limit, we can analyze the asymptotic behaviors of the networks. Recently, this asymptotic analysis is referred to as the mean field theory of deep networks, and we follow the previously reported notations and terminology |
First, let us introduce the following variables for feedforward signal propagations: [MATH] and [MATH] In the context of deep learning, these variables have been utilized to explain the depth to which signals can sufficiently propagate. The variable [MATH] is the correlation between the activations for different input ... |
[EQUATION] [EQUATION] with [MATH] and [MATH] [MATH] ). We can generalize the theory to unnormalized data with [MATH] and [MATH] , just by substituting them into the recurrence relations. The notation [MATH] means integration over the standard Gaussian density. Here, the notation [MATH] represents the following integral... |
[MATH] with [MATH] The [MATH] is linked to the compositional kernel and utilized as the kernel of the Gaussian process Next, let us introduce variables for backpropagated signals: [MATH] and [MATH] . Note that they are defined not by averages but by the sums. They remain [MATH] because of [MATH] |
[MATH] is the correlation of backpropagated signals. To compute these quantities, the previous studies assumed the following: Assumption 1 |
Schoenholz et al. On the evaluation of the variables [MATH] and [MATH] one can use a different set of parameters, [MATH] for the forward chain ( ) and [MATH] for the backpropagated chain ( ), instead of using the same parameter set [MATH] in both chains. |
This assumption makes the dependence between [MATH] (or [MATH] ) and [MATH] , which share the same parameter set, very weak, and one can regard it as independent. It enables us to apply the central limit theorem to the backpropagated chain ( ). Thus, the previous studies |
derived the following recurrence relations ( [MATH] ): [EQUATION] [EQUATION] with [MATH] because of the linear outputs. The previous works confirmed excellent agreements between the above equations and experiments. In this study, we also adopt the above assumption and use the recurrence relations. |
The variables ( [MATH] ) depend only on the variance parameters [MATH] and [MATH] , not on the unit indices. In that sense, they are referred to as macroscopic variables (a.k.a. order parameters in statistical physics). The recurrence relations for the macroscopic variables simply require [MATH] iterations of one- and ... |
Fundamental FIM statistics Here, we report mathematical findings that the mean, variance, and maximum of eigenvalues of the FIM ( ) are explicitly expressed by using macroscopic variables. Our theorems are universal for networks ranging in size from shallow ( [MATH] ) to arbitrarily deep ( [MATH] ) with various activat... |
3.1 Mean of eigenvalues The FIM is a [MATH] matrix, where [MATH] represents the total number of parameters. First, we compute the arithmetic mean of the FIM’s eigenvalues as [MATH] . We find a hidden relation between the macroscopic variables and the statistics of FIM: |
Theorem 1 Suppose that Assumption 1 holds. In the limit of [MATH] the mean of the FIM’s eigenvalues is given by [EQUATION] where [MATH] The macroscopic variables [MATH] and [MATH] can be computed recursively, and notably [MATH] is [MATH] |
This is obtained from a relation [MATH] (detailed in Supplementary Material A.1). The coefficient [MATH] is a constant not depending on [MATH] , so it is [MATH] . It is easily computed by [MATH] iterations of the layer-wise recurrence relations ( ) and ( 11 ). |
Because the FIM is a positive semi-definite matrix and its eigenvalues are non-negative, this theorem means that most of the eigenvalues asymptotically approach zero when [MATH] is large. Recall that the FIM determines the local geometry of the parameter space. The theorem suggests that the network output remains almos... |
Furthermore, by using Markov’s inequality, we can prove that the number of larger eigenvalues is limited, as follows: Corollary 2 |
Let us denote the number of eigenvalues satisfying [MATH] by [MATH] and suppose that Assumption 1 holds. For a constant [MATH] [MATH] holds in the limit of [MATH] |
The proof is shown in Supplementary Material A.2. When [MATH] is sufficiently small, we have a trivial upper bound [MATH] and the number of non-zero eigenvalue is limited. The corollary clarifies that even when [MATH] becomes large, the number of eigenvalues whose values are [MATH] is [MATH] at most, and still much sma... |
3.2 Variance of eigenvalues Next, let us consider the second moment [MATH] We now demonstrate that [MATH] can be computed from the macroscopic variables: |
Theorem 3 Suppose that Assumption 1 holds. In the limit of [MATH] , the second moment of the FIM’s eigenvalues is [EQUATION] The macroscopic variables |
[MATH] and [MATH] can be computed recursively, and [MATH] is [MATH] The proof is shown in Supplementary Material A.3. From Theorems 1 and 3, we can conclude that the variance of the eigenvalue distribution, [MATH] , is [MATH] . Because the mean [MATH] is [MATH] and most eigenvalues are close to zero, this result means ... |
3.3 Maximum eigenvalue As we have seen so far, the mean of the eigenvalues is [MATH] , and the variance is [MATH] Therefore, we can expect that at least one of the eigenvalues must be huge. Actually, we can show that the maximum eigenvalue (that is, the spectral norm of the FIM) increases in the order of [MATH] as foll... |
Theorem 4 Suppose that Assumption 1 holds. In the limit of [MATH] , the maximum eigenvalue of the FIM is [EQUATION] The [MATH] is derived from the dual matrix [MATH] (detailed in Supplemental Material A.4). If we take the limit [MATH] , we can characterize the quantity [MATH] by the maximum eigenvalue as [MATH] Note th... |
This theorem suggests that the network output changes dramatically with a perturbation of the parameters in certain dimensions and that the local shape of the loss landscape is strongly distorted in that direction. Here, note that [MATH] is proportional to [MATH] , which is the summation over [MATH] terms. This means t... |
We confirmed the agreement between our theory and numerical experiments, as shown in Fig. 1. Three types of deep networks with parameters ) were investigated: tanh, ReLU, and linear activations ( [MATH] [MATH] ). The input samples were generated using i.i.d. Gaussian samples, and [MATH] When [MATH] , we calculated the ... |
Connections to learning strategies Here, we show some applications that demonstrate how our universal theory on the FIM can potentially enrich deep learning theories. It enables us to quantitatively measure the behaviors of learning strategies as follows. |
4.1 The Fisher-Rao norm Recently, Liang et al. proposed the Fisher-Rao norm for a capacity measure of generalization ability: [EQUATION] |
where [MATH] represents weight parameters. They reported that this norm has several desirable properties to explain the high generalization capability of DNNs. In deep linear networks, its generalization capacity (Rademacher complexity) is upper bounded by the norm. In deep ReLU networks, the Fisher-Rao norm serves as ... |
and the spectral norm The Fisher-Rao norm is also motivated by information geometry, and invariant under node-wise linear rescaling in ReLU networks. This is a desirable property to connect capacity measures with flatness induced by the rescaling |
Here, to obtain a typical evaluation of the norm, we define the average over possible parameters with fixed variances ( [MATH] ) by [MATH] , which leads to the following theorem: |
Theorem 5 Suppose that Assumption 1 holds. In the limit of [MATH] , the Fisher-Rao norm of DNNs satisfies [EQUATION] where [MATH] . Equality holds in a network with a uniform width [MATH] , and then we have [MATH] |
The proof is shown in Supplementary Material A.6. Although what we can evaluate is only the average of the norm, it can be quantified by [MATH] This guarantees that the norm is independent of the network width in the limit of [MATH] , which was empirically conjectured by |
Recently, Smith and Le argued that the Bayesian factor composed of the Hessian of the loss function, whose special case is the FIM, is related to the generalization. Similar analysis to the above theorem may enable us to quantitatively understand the relation between the statistics of the FIM and the indicators to meas... |
4.2 Learning rate for convergence Consider the steepest gradient descent method in a batch regime. Its update rule is given by [EQUATION] |
where [MATH] is a constant learning rate. We have added a momentum term with a coefficient [MATH] because it is widely used in training deep networks. Assume that the squared loss function [MATH] of Eq. ( ) has a global minimum [MATH] achieving the zero training error [MATH] Then, the FIM’s maximum eigenvalue is domina... |
Lemma 6 A learning rate satisfying [MATH] is necessary for the steepest gradient method to converge to the global minimum [MATH] |
The proof is given by the expansion around the minimum, i.e., [MATH] (detailed in Supplementary Material A.7). This lemma is a generalization of LeCun et al. , which proved the case of [MATH] Let us refer to [MATH] as the critical learning rate. When [MATH] , the gradient method never converges to the global minimum. T... |
also claimed that [MATH] is the best choice for fastest convergence around the minimum. Although we focus on the batch regime, the eigenvalues also determine the bound of the gradient norms and the convergence of learning in the online regime |
Then, combining Lemma with Theorem leads to the following: Theorem 7 Suppose that Assumption 1 holds. Let a global minimum [MATH] be ) and satisfying [MATH] . In the limit of [MATH] , the gradient method never converges to [MATH] when |
[EQUATION] Theorem quantitatively reveals that, the wider the network becomes, the smaller the learning rate we need to set. In addition, [MATH] is the sum over [MATH] constant positive terms, so a deeper network requires a finer setting of the learning rate and it will make the optimization more difficult. In contrast... |
. We thus expect there to be a trade-off between trainability and expressive power. To confirm the effectiveness of Theorem , we performed several experiments. As shown in Fig. 2, we exhaustively searched training losses while changing [MATH] and [MATH] , and found that the theoretical estimation coincides well with th... |
The left column of Fig. 2 shows the results of training on artificial data. We generated training samples [MATH] in the Gaussian manner ( [MATH] ) and teacher signals [MATH] by the teacher network with a true parameter set [MATH] satisfying Eq. ( ). We used the gradient method ( 19 ) with [MATH] and trained the DNNs fo... |
We performed similar experiments on benchmark datasets and found that the theory can estimate the appropriate learning rates. The results on MNIST are shown in the right column of Fig. 2. As shown in Supplementary Material C.2, the results of training on CIFAR-10 were almost the same as those of MINIST. We used stochas... |
recommended to use for achieving high expressive power and trainability. Note that the variances [MATH] may change from the initialization to the global minimum, and the conditions of the global minimum in Theorem do not hold in general. Nevertheless, the learning rates estimated by Theorem explained the experiments we... |
Theoretical estimations of learning rates in deep networks have so far been limited; such gradients as AdaGrad and Adam also require heuristically determined hyper-parameters for learning rates. Extending our framework would be beneficial in guessing learning rates to prevent the gradient update from exploding. |
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