l1loss

L₁ loss for regression tasks

Since R2021b

Syntax

loss = l1loss(Y,targets)

loss = l1loss(Y,targets,weights)

loss = l1loss(___,DataFormat=FMT)

loss = l1loss(___,Name=Value)

Description

The L₁ loss operation computes the L₁ loss given network predictions and target values. When the Reduction option is "sum" and the NormalizationFactor option is "batch-size", the computed value is known as the mean absolute error (MAE).

The l1loss function calculates the L₁ loss using dlarray data. Using dlarray objects makes working with high dimensional data easier by allowing you to label the dimensions. For example, you can label which dimensions correspond to spatial, time, channel, and batch dimensions using the "S", "T", "C", and "B" labels, respectively. For unspecified and other dimensions, use the "U" label. For dlarray object functions that operate over particular dimensions, you can specify the dimension labels by formatting the dlarray object directly, or by using the DataFormat option.

loss = l1loss(Y,targets) computes the MAE loss for the predictions Y and the target values targets. The input Y must be a formatted dlarray. The output loss is an unformatted dlarray scalar.

loss = l1loss(Y,targets,weights) computes the weighted L₁ loss using the weight values weights. The output loss is an unformatted dlarray scalar.

loss = l1loss(___,DataFormat=FMT) computes the loss for the unformatted dlarray object Y and the target values with the format specified by FMT. Use this syntax with any of the input arguments in previous syntaxes.

loss = l1loss(___,Name=Value) specifies additional options using one or more name-value arguments. For example, l1loss(Y,targets,Reduction="none") computes the L₁ loss without reducing the output to a scalar.

Examples

Mean Absolute Error Loss

Create an array of predictions for 12 observations over 10 responses.

numResponses = 10;
numObservations = 12;

Y = rand(numResponses,numObservations);
dlY = dlarray(Y,'CB');

View the size and format of the predictions.

size(dlY)

ans = 1×2

    10    12

dims(dlY)

ans = 
'CB'

Create an array of random targets.

targets = rand(numResponses,numObservations);

View the size of the targets.

size(targets)

ans = 1×2

    10    12

Compute the mean absolute error (MAE) loss between the predictions and the targets using the l1loss function.

loss = l1loss(dlY,targets)

loss = 
  1x1 dlarray

    3.1679

Masked Mean Absolute Error for Padded Sequences

Create arrays of predictions and targets for 12 sequences of varying lengths over 10 responses.

numResponses = 10;
numObservations = 12;
maxSequenceLength = 15;

sequenceLengths = randi(maxSequenceLength,[1 numObservations]);

Y = cell(numObservations,1);
targets = cell(numObservations,1);

for i = 1:numObservations
    Y{i} = rand(numResponses,sequenceLengths(i));
    targets{i} = rand(numResponses,sequenceLengths(i));
end

View the cell arrays of predictions and targets.

Y

Y=12×1 cell array
    {10x13 double}
    {10x14 double}
    {10x2  double}
    {10x14 double}
    {10x10 double}
    {10x2  double}
    {10x5  double}
    {10x9  double}
    {10x15 double}
    {10x15 double}
    {10x3  double}
    {10x15 double}

targets

targets=12×1 cell array
    {10x13 double}
    {10x14 double}
    {10x2  double}
    {10x14 double}
    {10x10 double}
    {10x2  double}
    {10x5  double}
    {10x9  double}
    {10x15 double}
    {10x15 double}
    {10x3  double}
    {10x15 double}

Pad the prediction and target sequences in the second dimension using the padsequences function and also return the corresponding mask.

[Y,mask] = padsequences(Y,2);
targets = padsequences(targets,2);

Convert the padded sequences to dlarray with the format "CTB" (channel, time, batch). Because formatted dlarray objects automatically permute the dimensions of the underlying data, keep the order consistent by also converting the targets and mask to formatted dlarray objects with the format "CTB" (channel, batch, time).

dlY = dlarray(Y,"CTB");
targets = dlarray(targets,"CTB");
mask = dlarray(mask,"CTB");

View the sizes of the prediction scores, targets, and mask.

size(dlY)

ans = 1×3

    10    12    15

size(targets)

ans = 1×3

    10    12    15

size(mask)

ans = 1×3

    10    12    15

Compute the mean absolute error (MAE) between the predictions and the targets. To prevent the loss values calculated from padding from contributing to the loss, set the Mask option to the mask returned by the padsequences function.

loss = l1loss(dlY,targets,Mask=mask)

loss = 
  1x1 dlarray

   32.6172

Input Arguments

Y — Predictions

dlarray | numeric array

Predictions, specified as a formatted dlarray, an unformatted dlarray, or a numeric array. When Y is not a formatted dlarray, you must specify the dimension format using the DataFormat option.

If Y is a numeric array, targets must be a dlarray.

targets — Target responses

dlarray | numeric array

Target responses, specified as a formatted or unformatted dlarray or a numeric array.

The size of each dimension of targets must match the size of the corresponding dimension of Y.

If targets is a formatted dlarray, then its format must be the same as the format of Y, or the same as DataFormat if Y is unformatted.

If targets is an unformatted dlarray or a numeric array, then the function applies the format of Y or the value of DataFormat to targets.

Tip

Formatted dlarray objects automatically permute the dimensions of the underlying data to have order "S" (spatial), "C" (channel), "B" (batch), "T" (time), then "U" (unspecified). To ensure that the dimensions of Y and targets are consistent, when Y is a formatted dlarray, also specify targets as a formatted dlarray.

weights — Weights

dlarray | numeric array

Weights, specified as a formatted or unformatted dlarray or a numeric array.

If weights is a vector and Y has two or more nonsingleton dimensions, then weights must be a formatted dlarray, where the dimension label of the nonsingleton dimension is either "C" (channel) or "B" (batch) and has a size that matches the size of the corresponding dimension in Y.

If weights is a formatted dlarray with two or more nonsingleton dimensions, then its format must match the format of Y.

If weights is not a formatted dlarray and has two or more nonsingleton dimensions, then its size must match the size of Y and the function uses the same format as Y. Alternatively, to specify the weights format, use the WeightsFormat option.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: loss = l1loss(Y,targets,Reduction="none") specifies to compute the L₁ loss without reducing the output to a scalar

Mask — Mask indicating which elements to include for loss computation

dlarray | logical array | numeric array

Mask indicating which elements to include for loss computation, specified as a dlarray object, a logical array, or a numeric array with the same size as Y.

The function includes and excludes elements of the input data for loss computation when the corresponding value in the mask is 1 and 0, respectively.

If Mask is a formatted dlarray object, then its format must match that of Y. If Mask is not a formatted dlarray object, then the function uses the same format as Y.

If you specify the DataFormat option, then the function also uses the specified format for the mask.

The size of each dimension of Mask must match the size of the corresponding dimension in Y. The default value is a logical array of ones.

Tip

Formatted dlarray objects automatically permute the dimensions of the underlying data to have this order: "S" (spatial), "C" (channel), "B" (batch), "T" (time), and "U" (unspecified). For example, dlarray objects automatically permute the dimensions of data with format "TSCSBS" to have format "SSSCBT".

To ensure that the dimensions of Y and the mask are consistent, when Y is a formatted dlarray, also specify the mask as a formatted dlarray.

Reduction — Mode for reducing array of loss values

"sum" (default) | "none"

Mode for reducing the array of loss values, specified as one of the following:

• "sum" — Sum all of the elements in the array of loss values. In this case, the output loss is scalar.

• "none" — Do not reduce the array of loss values. In this case, the output loss is an unformatted dlarray object with the same size as Y.

NormalizationFactor — Divisor for normalizing reduced loss

"batch-size" (default) | "all-elements" | "mask-included" | "none"

Divisor for normalizing the reduced loss when Reduction is "sum", specified as one of the following:

• "batch-size" — Normalize the loss by dividing it by the number of observations in Y.

• "all-elements" — Normalize the loss by dividing it by the number of elements of Y.

• "mask-included" — Normalize the loss by dividing the loss values by the number of included elements specified by the mask for each observation independently. To use this option, you must specify a mask using the Mask option.

• "none" — Do not normalize the loss.

DataFormat — Dimension order of unformatted data

character vector | string scalar

Dimension order of unformatted input data, specified as a character vector or string scalar FMT that provides a label for each dimension of the data.

When you specify the format of a dlarray object, each character provides a label for each dimension of the data and must be one of these options:

• "S" — Spatial

• "C" — Channel

• "B" — Batch (for example, samples and observations)

• "T" — Time (for example, time steps of sequences)

• "U" — Unspecified

You can specify multiple dimensions labeled "S" or "U". You can use the labels "C", "B", and "T" at most once.

You must specify DataFormat when the input data is not a formatted dlarray.

Data Types: char | string

WeightsFormat — Dimension order of weights

character vector | string scalar

Dimension order of the weights, specified as a character vector or string scalar that provides a label for each dimension of the weights.

When you specify the format of a dlarray object, each character provides a label for each dimension of the data and must be one of these options:

• "S" — Spatial

• "C" — Channel

• "B" — Batch (for example, samples and observations)

• "T" — Time (for example, time steps of sequences)

• "U" — Unspecified

You can specify multiple dimensions labeled "S" or "U". You can use the labels "C", "B", and "T" at most once.

You must specify WeightsFormat when weights is a numeric vector and Y has two or more nonsingleton dimensions.

If weights is not a vector, or both weights and Y are vectors, then default value of WeightsFormat is the same as the format of Y.

Data Types: char | string

Output Arguments

loss — L₁ loss

dlarray

L₁ loss, returned as an unformatted dlarray. The output loss is an unformatted dlarray with the same underlying data type as the input Y.

The size of loss depends on the Reduction option.

Algorithms

L1 Loss

The L₁ loss operation computes the L₁ loss given network predictions and target values. When the Reduction option is "sum" and the NormalizationFactor option is "batch-size", the computed value is known as the mean absolute error (MAE).

For each element Y_j of the input, the l1loss function computes the corresponding element-wise loss values using

$${\text{loss}}_j=\left|Y_j-T_j\right|,$$

where Y_j is a predicted value and T_j is the corresponding target value.

To reduce the loss values to a scalar, the function then reduces the element-wise loss using the formula

$$\text{loss}=\frac{1}{N}{\displaystyle \sum _j{m}_j{w}_j{\text{loss}}_j,}$$

where N is the normalization factor, m_j is the mask value for element j, and w_j is the weight value for element j.

If you do not opt to reduce the loss, then the function applies the mask and the weights to the loss values directly:

$${\text{loss}}_j^{*}=m_j{w}_j{\text{loss}}_j$$

Extended Capabilities

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

• When at least one of the following input arguments is a gpuArray or a dlarray with underlying data of type gpuArray, this function runs on the GPU:

  • Y

  • targets

  • weights

  • Mask

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2021b