dljacobian

Jacobian matrix deep learning operation

Since R2024b

Syntax

jac = dljacobian(u,x,dim)

jac = dljacobian(u,x,dim,EnableHigherDerivatives=tf)

Description

The Jacobian deep learning operation returns the Jacobian matrix for neural network and model function outputs with respect to the specified input data and operation dimension.

jac = dljacobian(u,x,dim) returns the Jacobian matrix for the neural network outputs u with respect to the data x for the specified operation dimension.

jac = dljacobian(u,x,dim,EnableHigherDerivatives=tf) also specifies whether to enable higher derivatives by tracing the backward pass.

Examples

Evaluate Jacobian of Deep Learning Data

Create a neural network.

inputSize = [16 16 3];
numOutputChannels = 5;

layers = [
    imageInputLayer(inputSize)
    convolution2dLayer(3,64)
    reluLayer
    fullyConnectedLayer(numOutputChannels)
    softmaxLayer];

net = dlnetwork(layers);

Load the training data. For the purposes of this example, generate some random data.

numObservations = 128;
X = rand([inputSize numObservations]);
X = dlarray(X,"SSCB");

T = rand([numOutputChannels numObservations]);
T = dlarray(T,"CB");

Define a model loss function that takes the network and data as input and returns the loss, gradients of the loss with respect to the learnable parameters, and the Jacobian of the predictions with respect to the input data.

function [loss,gradients,jac] = modelLoss(net,X,T)

Y = forward(net,X);
loss = l1loss(Y,T);

X = stripdims(X);
Y = stripdims(Y);

jac = dljacobian(Y,X,1);
gradients = dlgradient(loss,net.Learnables);

end

Evaluate the model loss function using the dlfeval function.

[loss,gradients,jac] = dlfeval(@modelLoss,net,X,T);

View the size of the Jacobian.

size(jac)

ans = 1×5

     5    16    16     3   128

Input Arguments

u — Input

traced dlarray matrix

Input, specified as a traced dlarray matrix.

When the software evaluates a function with automatic differentiation enabled, the software traces the input dlarray objects. These are some contexts where the software traces dlarray objects:

• Inside loss functions that the trainnet function evaluates

• Inside forward functions that custom layers evaluate

• Inside model and model loss functions that the dlfeval function evaluates

The sizes of the dimensions not specified by the dim argument must match.

x — Input

traced dlarray object

Input, specified as a traced dlarray object.

When the software evaluates a function with automatic differentiation enabled, the software traces the input dlarray objects. These are some contexts where the software traces dlarray objects:

• Inside loss functions that the trainnet function evaluates

• Inside forward functions that custom layers evaluate

• Inside model and model loss functions that the dlfeval function evaluates

The sizes of the dimensions not specified by the dim argument must match.

dim — Operation dimension

positive integer

Operation dimension of u, specified as a positive integer.

The dljacobian function treats the remaining dimensions of the data as independent batch dimensions.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

tf — Flag to enable higher-order derivatives

true or 1 (default) | false or 0

Flag to enable higher-order derivatives, specified as one of the following:

• true — Enable higher-order derivatives. Trace the backward pass so that the returned gradients can be used in further computations for subsequent calls to functions that compute derivatives using automatic differentiation (for example, dlgradient, dljacobian, dldivergence, and dllaplacian).

• false — Disable higher-order derivatives. Do not trace the backward pass. Use this option when you need to compute first-order derivatives only as this is usually quicker and requires less memory.

Output Arguments

jac — Jacobian matrix

unformatted dlarray object

Jacobian matrix, returned as an unformatted dlarray object.

The layout of jac depends on dim and the size of u.

If size(u,dim) == 1, then jac is a matrix, and:

• If dim is 1, then jac(j,k) corresponds to $$\nabla u_{(j,k)}=\frac{\partial u_k}{\partial x_{(j,k)}}$$ and u_k corresponds to u(:,k).

• If dim is 2, then jac(k,j) corresponds to $${(\nabla u)}_{(k,j)}=\frac{\partial u_k}{\partial x_{(k,j)}}$$ and u_k corresponds to u(k,:).

Otherwise, if size(u,dim) > 1, then jac is a 3-D array, and:

• If dim is 1, then jac(i,j,k) corresponds to $${(\nabla u)}_{(i,j,k)}=\frac{\partial u_{(k,i)}}{\partial x_{(j,k)}}$$.

• If dim is 2, then jac(i,k,j) corresponds to $${(\nabla u)}_{(i,k,j)}=\frac{\partial u_{(k,i)}}{\partial x_{(k,j)}}$$.

Version History

Introduced in R2024b