Amari, S. Natural Gradient Works E#ciently in Learning. Neural Computation 10, 1998  Home/Search   Document Not in Database   Summary   Related Articles   Check

....eqs. 21) and (32) Here, the underlying mechanism can be recognized as the classic Newton technique in which the gradient is left multiplied by the inverse of the Hessian for it to point in the best (in a certain sense) direction. Thus, it is likely that the natural gradient approach of Amari [20] would result in similar algorithms. We note that on line algorithm takes its particular simple form thanks to an approximation which is only valid when the model holds. When this is not the case, the algorithm is still well behaved because the gradient is rectified by a matrix which is always ....

Shun-Ichi Amari, "Natural gradient works e#ciently in learning", Neural Computation, vol. 10, pp. 251--276, 1998.

....y = Bx; thus it actually characterizes the first order variation of the contrast function itself. It is of course possible to relate to a regular gradient with respect to B if y = Bx. Elementary calculus yields #B . 29) The notion of natural gradient was independently introduced by Amari [2]. It is distinct in general from the relative gradient: the latter is defined in any continuous group of transformation while the former is defined in any smooth statistical model. However, for the BSS model which, as a statistical transformation model combines both features, the two ideas yield ....

S.-I. Amari. Natural gradient works e#ciently in learning. Neural Computation, 10:251--276, 1998.

....and network architecture guarantee local minima free error surfaces and the important contribution of [13] showing that the classical XOR problem has no real local minima. Furthermore, the fundamental and innovative approach founded on the natural gradient , introduced by Amari (see f.i. 1] [2], 17] was proved to be a powerful stochastic method to solve the critical problem of escaping from plateaus of the error surface, thereby ensuring an e ective steepest descent in any situation. Since algorithms based on the natural gradient have locally a QuasiNewtonian (QN) behaviour, ....

S.Amari, Natural gradient works eciently in lea rning, Neural Computation 10 (1998), 251-276.

....2 norm in d(I1 , I2 ; U) allows for this equality. Grassmann and similar manifolds have been used in the literature to derive e#ective algorithms but in di#erent contexts; see e.g. 18] for subspace tracking, 13] for motion and structure estimation, 15] for independent component analysis, and [1, 6] for neural network learning algorithms. gradient process has to account for its intrinsic geometry. Deterministic gradients such as NewtonRaphson method, on such manifolds with orthogonality constraints have been studied in [5] We will start by describing a deterministic gradient process (of F ....

....As shown in these examples and the ones in Figs. 2 and 3, the proposed algorithm converges quickly under di#erent kinds of conditions. We attribute this mainly to the fact that by taking into the geometry of the manifold into account, the gradient flow provides the most e#cient way for updating [1]. In fact, Amari [1] has shown that such a dynamical system is Fisher e#cient. Obviously, the e#ectiveness of the proposed algorithm depends on the choice of parameters; through experiments we have found that it works for a wide of parameter values. This can be attributed to the update rules in ....

[Article contains additional citation context not shown here]

S. Amari. Natural gradient works e#ciently in learning. Neural Computation, 10:251-276, 1998.

....t = # g W (t) 32) Here, t is the index for iteration and # t is a learning rate. 3.2. Removal of the Inverse Matrix In Equation (28) a matrix inverse and transpose W T appears. The matrix inversion and transposition can be removed by using a natural or relative gradient [1], 4] By considering (15) we multiply CW W . Then, we have #I (CW W ) dy W. 33) An important next step is how to evaluate the core of the integrand of (33) It is a key to observe qg # (q p ) g ## (1)p g # (1) g ## (1) q o(1) 34) ....

S. Amari, Natural gradient works e#ciently in learning, Neural Computation, vol. 10, pp. 252276, 1998.

....on the underlying structure of the policy. We make the connection to policy iteration by showing that the natural gradient is moving toward choosing a greedy optimal action. We then analyze the performance of the natural gradient in both simple and complicated MDPs. Consistent with Amari s ndings [1], our work suggests that the plateau phenomenon might not be as severe using this method. 2 A Natural Gradient A nite MDP is a tuple (S; s 0 ; A; R; P ) where: S is nite set of states, s 0 is a start state, A is a nite set of actions, R is a reward function R : S A [0; Rmax ] and P is the ....

....the constraint that the squared length jd j is held to a small constant. This squared length is de ned with respect to some positive de nite matrix G( ie jd j ij G ij ( d i d j = d G( d (using vector notation) The steepest descent direction is then given by G r ( [1]. Standard gradient descent follows the direction r ( which is the steepest descent under the assumption that G( is the identity matrix, I . However, this as hoc choice of a metric is not necessarily appropriate. As suggested by Amari [1] it is better to de ne a metric based not on the choice ....

[Article contains additional citation context not shown here]

S. Amari. Natural gradient works eciently in learning. Neural Computation, 10(2):251276, 1998.

....The ultimate test of the assumption is, of course, in the applications. When updating the model vectors to decrease their distance from the data vector it is essential to note that the new metric is non Euclidean. Hence, the direction of the steepest descent is given by the natural gradient [6]. It is straightforward to show that when the new metric and the natural gradient are used in the SOM algorithm, its update rule becomes the usual update rule of the Euclidean SOM, i.e. m j (t 1) m j (t) hwj (t) x(t) m j (t) 4) Here hwj is the neighborhood function, a decreasing ....

S.-I. Amari, \Natural Gradient Works Eciently in Learning," Neural Computation, vol. 10, pp. 251-276, 1998.

....step, each model vector m i (t) is updated in the direction where the distance to x(t) decreases most rapidly. In the Euclidean metric the direction is that of the negative gradient, 2(m i (t) x(t) but in the more general Riemannian metric the direction is oposite to the natural gradient [7] J(x) 1 m i (t) d 2 L (x; m i (t) 7) Using again the local approximation this becomes J(x) 1 2J(x) m i (t) x) 2(m i (t) x) 8) which is just the gradient of the Euclidean metric. Thus the model vectors will be adapted with the familiar rule (5) 3.2 Probability Estimation The ....

S.-I. Amari, \Natural gradient works eciently in learning," Neural Computation, vol. 10, pp. 251-276, 1998.

....current data y k = W k x and select suitable model (e.g. in the case of the EGLD make the choice between the GLD and the GBD) 2. Estimate parameters for the model by using the method of moments and calculate the scores (y k ) 3. Calculate the separating matrix W k 1 using a gradient algorithm [1, 7] or xed point (13) algorithm. The stability of optimization algorithms in source distribution adaptive maximum likelihood approach remains an open problem. The complexity of the EGLD and the Pearson system implies that analytical results are hard to obtain. However, stability problems are seldom ....

S.-I. Amari, \Natural Gradient Works Eciently in Learning," Neural Computation, vol. 10, pp. 251-276, 1998.

....= W p (t 1) W p (t) t dl dW p = t [I W 0 (t) T (y(t) y T (t p) 28) The algorithm (28) requires the calculation of inverse, I W 0 (t) T , so the computational complexity is relatively high. Alternatively, we can use the technique that was employed in the natural gradient [1]. We de ne a modi ed di erential matrix dV p by dV p = I W 0 ) 1 dW p : 29) Then (27) becomes dy(t) L X p=0 dV p y(t p) 30) With this modi ed di erential matrix, we have V p (t) t dl dV p = y(t) y T (t p) 31) Using the relation (29) the learning algorithm for W p is ....

S. Amari. Natural gradient works eciently in learning. Neural Computation, 10(2):251-276, 1998.

....o (x) was neglected since it does not depend on A. The loss function (4) can also be obtained from nonlinear infomax, maximum likelihood estimation, and mutual information minimization. Popular ICA algorithms were derived from the minimization of the loss function (4) using the natural gradient [1]. Many natural signals have non vanishing temporal correlations. These temporal structure of sources allows us to perform BSS using second order statistics only, in contrast to the conventional ICA method. Along this line, several methods have been developed [13, 7, 5, 2, 10] where most of them ....

....PI = 1 2(n 1) n X i=1 ( n X k=1 jg ik j 2 max j jg ij j 2 1 n X k=1 jg ki j 2 max j jg ji j 2 1 ) 31) where g ij is the (i; j) element of the global system matrix G = WA. We have compared our method to the conventional ICA algorithm (using the natural gradient) [1, 6] with the Laplacian prior (which results in signum nonlinear function) In both algorithms, the demixing matrix W was initialized as the identity matrix. The learning rates = 005 and = 01 were used for our method and the conventional ICA algorithm, respectively. Figure 1 shows the evolution ....

S. Amari. Natural gradient works eciently in learning. Neural Computation, 10(2):251-276, 1998.

....an orthogonal coordinate system, the conventional gradient gives the steepest descent direction. However, if a parameter space is a curved manifold (Riemannian space) an orthonormal linear coordinate system does not exist and the conventional gradient does not give the steepest descent direction [1]. Recently the natural gradient was proposed by Amari [1] and was shown to be ecient in on line learning. See [1] for more details of the natural gradient. Note that the relative gradient developed independently by Cardoso and Laheld [12] is identical to the natural gradient in the context of ICA. ....

....gives the steepest descent direction. However, if a parameter space is a curved manifold (Riemannian space) an orthonormal linear coordinate system does not exist and the conventional gradient does not give the steepest descent direction [1] Recently the natural gradient was proposed by Amari [1] and was shown to be ecient in on line learning. See [1] for more details of the natural gradient. Note that the relative gradient developed independently by Cardoso and Laheld [12] is identical to the natural gradient in the context of ICA. In this section, we brie y review two natural gradient ....

[Article contains additional citation context not shown here]

S. Amari. Natural gradient works eciently in learning. Neural Computation, 10(2):251-276, Feb. 1998.

....an orthogonal coordinate system, the conventional gradient gives the steepest descent direction. However, if a parameter space is acurved manifold (Riemannian space) an orthonormal linear coordinate system does not exist and the conventional gradient does not give the steepest descent direction [1]. Recently the natural gradient was proposed by Amari [1] and was shown to be ecient in on line learning. See [1] for more details of the natural gradient. Note that the relative gradient developed independently by Cardoso and Laheld [12] is identical to the natural gradient in the context of ICA. ....

....gives the steepest descent direction. However, if a parameter space is acurved manifold (Riemannian space) an orthonormal linear coordinate system does not exist and the conventional gradient does not give the steepest descent direction [1] Recently the natural gradient was proposed by Amari [1] and was shown to be ecient in on line learning. See [1] for more details of the natural gradient. Note that the relative gradient developed independently by Cardoso and Laheld [12] is identical to the natural gradient in the context of ICA. In this section, we brie y review two natural gradient ....

[Article contains additional citation context not shown here]

S. Amari. Natural gradientworks eciently in learning. ###### ###########, 10(2):251-276, Feb. 1998.

....also considered in Lewicki Sejnowski work on learning overcomplete (underdetermined) representation [6] 3.2. Natural Gradient Adaptation Algorithm The natural gradient was shown to be ecient in on line learning and to nd the steepest direction when the parameter space is Riemannian manifold [7]. The natural gradient learning was successfully applied to the task of blind source separation and multichannel blind deconvolution [8, 1, 9, 10] However, all these methods were restricted to the case of complete noise free mixtures. Here we derive the natural gradient algorithm for estimating ....

....n) fA 2 IR m n jrank(A) ng: 23) For A 2 Gl(m; n) there exits an orthogonal matrix Q 2 IR m m such that A = Q T A1 A2 ; 24) where A1 2 IR n n and A2 2 IR (m n) n . The Lie group structure of the parameter space is a key ingredient in the calculation of the natural gradient [7]. In similar manner as in [5] we introduce two operations on the manifold Gl(m; n) X Y = Q T X 1 Y 1 X 2 Y 1 Y 2 ; 25) and X # = Q T X 1 1 X 2X 1 ; 26) where X = Q T X 1 X 2 ; Y = Q T Y 1 Y 2 : 27) The operator denotes the multiplication and # ....

[Article contains additional citation context not shown here]

S. Amari, \Natural gradient works eciently in learning, " Neural Computation, vol. 10, no. 2, pp. 251-276, 1998.

No context found.

Amari, S. Natural Gradient Works E#ciently in Learning. Neural Computation 10, 1998

No context found.

Amari, S. Natural Gradient Works E#ciently in Learning. Neural Computation 10, 1998

No context found.

S. Amari. Natural gradient works e#ciently in learning. Neural Computation, 10:251--276, 1998.

No context found.

S. Amari. Natural gradient works e#ciently in learning. Neural Computation, 10(2):251--276, 1998.

No context found.

Amari, S. (1998). Natural gradient works e#ciently in learning. Neural Computation 10, 251--276.

No context found.

Amari, S. (1998). Natural gradient works e#ciently in learning. Neural Computation, 10(2):251--276.

No context found.

S. Amari, "Natural Gradient Works E#ciently in Learning," Neural Computation, vol. 10, pp. 251-276, 1998.

No context found.

Amari, S., 1998. Natural gradient works e#ciently in learning. Neural Computation 10, 251--276.

No context found.

S.-I. Amari, \Natural gradient works eciently in learning," Neural Computation, vol. 10, pp. 251-276, 1998.

No context found.

S. Amari. Natural gradient works eciently in learning. Neural Computation, 10(2):251{ 276, 1998.

No context found.

Shun-ichi Amari, Natural gradient works eciently in learning, Neural Computation 10 (1998), 251-276.

No context found.

S. Amari. Natural gradient works e#ciently in learning. Neural Computation, 10(2):251--276, 1998. amari98natural. html

No context found.

S. Amari. Natural gradient works e#ciently in learning. Neural Computation, 10:251-276, 1998.

No context found.

S.-I. Amari. "Natural gradient works e#ciently in learning," Neural Computation, vol. 10, 1998, pp. 251--276. 14

No context found.

S.-I. Amari. Natural gradient works eciently in learning. Neural Computation, 10(2):251276, 1998.

No context found.

S. Amari. Natural gradient works eciently in learning. Neural Computation, 10:251{ 276, 1998.

No context found.

S. Amari, Natural gradient works e#ciently in learning, Neural Computation 10 (1998) 251--276.

No context found.

Shun-ichi Amari. Natural gradient works e#ciently in learning. In Neural Computation, 10, pp. 251--276, 1998.

No context found.

S. Amari, \Natural Gradient Works Eciently in Learning", Neural Computation, Vol. 10, pp. 251-276, 1998.

No context found.

S. Amari, "Natural gradient works e#ciently in learning," Neural Computation, vol. 10, no. 2, pp. 251--276, 1998.

No context found.

S.-I. Amari 1998, "Natural gradient works e#ciently in learning," Neural Computation 10, 251--276.

No context found.

S. Amari, "Natural gradient works e ciently in learning, " Neural Comput., vol.10, pp.251-276, 1998.

No context found.

: S. Amari. Natural gradient works eciently in learning. Neural Computation, 10(2):251{

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC