Linear algebra ops

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cholesky function

keras.ops.cholesky(x, upper=False)

Computes the Cholesky decomposition of a positive semi-definite matrix.

Arguments

• x: Input tensor of shape (..., M, M).
• upper (bool): If True, returns the upper-triangular Cholesky factor. If False (default), returns the lower-triangular Cholesky factor.

Returns

A tensor of shape (..., M, M) representing the Cholesky factor of x.

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det function

keras.ops.det(x)

Computes the determinant of a square tensor.

Arguments

• x: Input tensor of shape (..., M, M).

Returns

A tensor of shape (...,) representing the determinant of x.

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eig function

keras.ops.eig(x)

Computes the eigenvalues and eigenvectors of a square matrix.

Arguments

• x: Input tensor of shape (..., M, M).

Returns

• A tuple of two tensors: a tensor of shape (..., M) containing eigenvalues and a tensor of shape (..., M, M) containing eigenvectors.

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eigh function

keras.ops.eigh(x)

Computes the eigenvalues and eigenvectors of a complex Hermitian.

Arguments

• x: Input tensor of shape (..., M, M).

Returns

• A tuple of two tensors: a tensor of shape (..., M) containing eigenvalues and a tensor of shape (..., M, M) containing eigenvectors.

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inv function

keras.ops.inv(x)

Computes the inverse of a square tensor.

Arguments

• x: Input tensor of shape (..., M, M).

Returns

A tensor of shape (..., M, M) representing the inverse of x.

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jvp function

keras.ops.jvp(fun, primals, tangents, has_aux=False)

Computes a (forward-mode) Jacobian-vector product of fun. Arguments

• fun: Function to be differentiated. Its arguments should be arrays, scalars, or standard Python containers of arrays or scalars. It should return an array, scalar, or standard Python container of arrays or scalars.
• primals: The primal values at which the Jacobian of fun should be evaluated. Should be either a tuple or a list of arguments, and its length should be equal to the number of positional parameters of fun.
• tangents: The tangent vector for which the Jacobian-vector product should be evaluated. Should be either a tuple or a list of tangents, with the same tree structure and array shapes as primals.
• has_aux: Optional, bool. Indicates whether fun returns a pair where the first element is considered the output of the mathematical function to be differentiated and the second element is auxiliary data. Default is False.

Returns

If has_aux is False, returns a (primals_out, tangents_out) pair, where primals_out is fun(*primals), and tangents_out is the Jacobian-vector product of fun evaluated at primals with tangents. The tangents_out value has the same Python tree structure and shapes as primals_out.

If has_aux is True, returns a (primals_out, tangents_out, aux) tuple where aux is the auxiliary data returned by fun.

Example

>>> from keras import ops
>>> a1, a2 = ops.convert_to_tensor(0.1), ops.convert_to_tensor(0.2)
>>> primals, tangents = ops.jvp(ops.sin, (a1,), (a2,))
>>> primals
0.09983342
>>> tangents
0.19900084

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logdet function

keras.ops.logdet(x)

Computes log of the determinant of a hermitian positive definite matrix.

Arguments

• x: Input matrix. It must 2D and square.

Returns

The natural log of the determinant of matrix.

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lstsq function

keras.ops.lstsq(a, b, rcond=None)

Return the least-squares solution to a linear matrix equation.

Computes the vector x that approximately solves the equation a @ x = b. The equation may be under-, well-, or over-determined (i.e., the number of linearly independent rows of a can be less than, equal to, or greater than its number of linearly independent columns). If a is square and of full rank, then x (but for round-off error) is the exact solution of the equation. Else, x minimizes the L2 norm of b - a * x.

If there are multiple minimizing solutions, the one with the smallest L2 norm is returned.

Arguments

• a: "Coefficient" matrix of shape (M, N).
• b: Ordinate or "dependent variable" values, of shape (M,) or (M, K). If b is two-dimensional, the least-squares solution is calculated for each of the K columns of b.
• rcond: Cut-off ratio for small singular values of a. For the purposes of rank determination, singular values are treated as zero if they are smaller than rcond times the largest singular value of a.

Returns

Tensor with shape (N,) or (N, K) containing the least-squares solutions.

NOTE: The output differs from numpy.linalg.lstsq. NumPy returns a tuple with four elements, the first of which being the least-squares solutions and the others being essentially never used. Keras only returns the first value. This is done both to ensure consistency across backends (which cannot be achieved for the other values) and to simplify the API.

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lu_factor function

keras.ops.lu_factor(x)

Computes the lower-upper decomposition of a square matrix.

Arguments

• x: A tensor of shape (..., M, M).

Returns

• A tuple of two tensors: a tensor of shape (..., M, M) containing the lower and upper triangular matrices and a tensor of shape (..., M) containing the pivots.

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norm function

keras.ops.norm(x, ord=None, axis=None, keepdims=False)

Matrix or vector norm.

This function is able to return one of eight different matrix norms, or one of an infinite number of vector norms (described below), depending on the value of the ord parameter.

Arguments

• x: Input tensor.
• ord: Order of the norm (see table under Notes). The default is None.
• axis: If axis is an integer, it specifies the axis of x along which to compute the vector norms. If axis is a 2-tuple, it specifies the axes that hold 2-D matrices, and the matrix norms of these matrices are computed.
• keepdims: If this is set to True, the axes which are reduced are left in the result as dimensions with size one.

Note: For values of ord < 1, the result is, strictly speaking, not a mathematical 'norm', but it may still be useful for various numerical purposes. The following norms can be calculated: - For matrices: - ord=None: Frobenius norm - ord="fro": Frobenius norm - ord="nuc": nuclear norm - ord=np.inf: max(sum(abs(x), axis=1)) - ord=-np.inf: min(sum(abs(x), axis=1)) - ord=0: not supported - ord=1: max(sum(abs(x), axis=0)) - ord=-1: min(sum(abs(x), axis=0)) - ord=2: 2-norm (largest sing. value) - ord=-2: smallest singular value - other: not supported - For vectors: - ord=None: 2-norm - ord="fro": not supported - ord="nuc": not supported - ord=np.inf: max(abs(x)) - ord=-np.inf: min(abs(x)) - ord=0: sum(x != 0) - ord=1: as below - ord=-1: as below - ord=2: as below - ord=-2: as below - other: sum(abs(x)**ord)**(1./ord)

Returns

Norm of the matrix or vector(s).

Example

>>> x = keras.ops.reshape(keras.ops.arange(9, dtype="float32") - 4, (3, 3))
>>> keras.ops.linalg.norm(x)
7.7459664

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qr function

keras.ops.qr(x, mode="reduced")

Computes the QR decomposition of a tensor.

Arguments

• x: Input tensor of shape (..., M, N).
• mode: A string specifying the mode of the QR decomposition.
  • 'reduced': Returns the reduced QR decomposition. (default)
  • 'complete': Returns the complete QR decomposition.

Returns

A tuple containing two tensors. The first tensor of shape (..., M, K) is the orthogonal matrix q and the second tensor of shape (..., K, N) is the upper triangular matrix r, where K = min(M, N).

Example

>>> x = keras.ops.convert_to_tensor([[1., 2.], [3., 4.], [5., 6.]])
>>> q, r = qr(x)
>>> print(q)
array([[-0.16903079  0.897085]
       [-0.5070925   0.2760267 ]
       [-0.8451542  -0.34503305]], shape=(3, 2), dtype=float32)

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solve function

keras.ops.solve(a, b)

Solves a linear system of equations given by a x = b.

Arguments

• a: A tensor of shape (..., M, M) representing the coefficients matrix.
• b: A tensor of shape (..., M) or (..., M, N) representing the right-hand side or "dependent variable" matrix.

Returns

A tensor of shape (..., M) or (..., M, N) representing the solution of the linear system. Returned shape is identical to b.

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solve_triangular function

keras.ops.solve_triangular(a, b, lower=False)

Solves a linear system of equations given by a x = b.

Arguments

• a: A tensor of shape (..., M, M) representing the coefficients matrix.
• b: A tensor of shape (..., M) or (..., M, N) representing the right-hand side or "dependent variable" matrix.

Returns

A tensor of shape (..., M) or (..., M, N) representing the solution of the linear system. Returned shape is identical to b.

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svd function

keras.ops.svd(x, full_matrices=True, compute_uv=True)

Computes the singular value decomposition of a matrix.

Arguments

• x: Input tensor of shape (..., M, N).

Returns

• A tuple of three tensors: a tensor of shape (..., M, M) containing the left singular vectors, a tensor of shape (..., M, N) containing the singular values and a tensor of shape (..., N, N) containing the right singular vectors.