dymad.modules.kernel¶

Functions

`inv_softplus`(y, dtype)	Inverse of softplus, for initialization.
`scaled_cdist`(X, Z, scale, p)	Pairwise distance \|\|X/scale - Z/scale\|\|^p with broadcasting-friendly scaling.

Classes

`KernelAbstract`(in_dim[, dtype])	Base interface for all kernels (scalar or operator-valued).
`KernelOpSeparable`(kernels, out_dim[, Ls, dtype])	Separable operator-valued kernel K(x,z) = sum_i k_i(x,z; ell) * B_i where B_i = L_i L_i^T is PSD and learnable.
`KernelOpTangent`(kernel, out_dim[, dtype])	Operator-valued kernel for vector fields on a manifold
`KernelOperatorValued`(in_dim, out_dim[, dtype])
`KernelOperatorValuedScalars`(kernels, out_dim)	Operator-valued kernel induced by scalar kernels Output shape: (..., N, Dy, M, Dy)
`KernelScDM`(in_dim[, eps_init, t_init, dtype])	Symmetric-normalized diffusion kernel via diffusion maps.
`KernelScExp`(in_dim[, lengthscale_init, dtype])	Scalar Exponential: k(x,z) = exp(-\|\|x - z\|\| / ell) Learnable positive lengthscale.
`KernelScRBF`(in_dim[, lengthscale_init, dtype])	Scalar RBF: k(x,z) = exp(-0.5 * \|\|x - z\|\|^2 / ell^2) Learnable positive lengthscale.
`KernelScalarValued`(in_dim[, dtype])

class dymad.modules.kernel.KernelAbstract(in_dim, dtype=None)¶

Bases: Module, ABC

Base interface for all kernels (scalar or operator-valued).

abstractmethod forward(X, Z=None)¶

Compute kernel between X (N,d) and Z (M,d).

If Z is None, compute K(X,X).

Returns:

(N, M) - Operator-valued kernels: (N, Dy, M, Dy)

Return type:

Scalar kernels

abstract property is_operator_valued: bool¶: True for operator-valued kernels; False for scalar kernels.

set_reference_data(Xref)¶

Prepare data-dependent structures from Xref (N,d). Must be differentiable if kernel params are learnable.

By default the kernel is data-independent and does nothing.

Return type:: None

class dymad.modules.kernel.KernelOpSeparable(kernels, out_dim, Ls=None, dtype=None)¶

Bases: KernelOperatorValuedScalars

Separable operator-valued kernel K(x,z) = sum_i k_i(x,z; ell) * B_i where B_i = L_i L_i^T is PSD and learnable. Output shape: (…, N, Dy, M, Dy)

forward(X, Z=None)¶

Compute kernel between X (N,d) and Z (M,d).

If Z is None, compute K(X,X).

Returns:

(N, M) - Operator-valued kernels: (N, Dy, M, Dy)

Return type:

Scalar kernels

class dymad.modules.kernel.KernelOpTangent(kernel, out_dim, dtype=None)¶

Bases: KernelOperatorValued

Operator-valued kernel for vector fields on a manifold

For manifold of intrinsic dimension d and ambient dimension Dy:

K(x,z) = k(x,z; ell) * T(x’) O(x’,z’) T(z’)^T

where O(x’,z’) = T(x’)^T T(z’) and T, of (Dy,d), are tangent basis vectors at x’ and z’, and the ‘ denotes the state part of the input (the first out_dim dimensions). k is a scalar kernel that includes both states and inputs.

Returns a factored representation of the kernel to stay in intrinsic dimension

k(x,z; ell) O(x,z), T(x), T(z)

of shapes: (…, d, M, d), (…, d, Dy), (M, d, Dy)

forward(X, Z=None)¶

Compute kernel between X (N,d) and Z (M,d).

If Z is None, compute K(X,X).

Returns:

(N, M) - Operator-valued kernels: (N, Dy, M, Dy)

Return type:

Scalar kernels

set_manifold(manifold)¶

Return type:: None

set_reference_data(Xref)¶

Prepare data-dependent structures from Xref (N,d). Must be differentiable if kernel params are learnable.

By default the kernel is data-independent and does nothing.

Return type:: None

class dymad.modules.kernel.KernelOperatorValued(in_dim, out_dim, dtype=None)¶

Bases: KernelAbstract, ABC

property is_operator_valued: bool¶: True for operator-valued kernels; False for scalar kernels.

class dymad.modules.kernel.KernelOperatorValuedScalars(kernels, out_dim, dtype=None)¶

Bases: KernelOperatorValued

Operator-valued kernel induced by scalar kernels Output shape: (…, N, Dy, M, Dy)

set_reference_data(Xref)¶

Prepare data-dependent structures from Xref (N,d). Must be differentiable if kernel params are learnable.

By default the kernel is data-independent and does nothing.

Return type:: None

class dymad.modules.kernel.KernelScDM(in_dim, eps_init=None, t_init=1.0, dtype=None)¶

Bases: KernelScalarValued

Symmetric-normalized diffusion kernel via diffusion maps.

Everything keeps autograd for eps and t.

property eps¶

forward(X, Z=None)¶

Compute kernel between X (N,d) and Z (M,d).

If Z is None, compute K(X,X).

Returns:

(N, M) - Operator-valued kernels: (N, Dy, M, Dy)

Return type:

Scalar kernels

set_reference_data(Xref)¶

Prepare data-dependent structures from Xref (N,d). Must be differentiable if kernel params are learnable.

By default the kernel is data-independent and does nothing.

Return type:: None

property t¶

class dymad.modules.kernel.KernelScExp(in_dim, lengthscale_init=None, dtype=None)¶

Bases: KernelScalarValued

Scalar Exponential: k(x,z) = exp(-||x - z|| / ell) Learnable positive lengthscale.

property ell¶

forward(X, Z=None)¶

Compute kernel between X (N,d) and Z (M,d).

If Z is None, compute K(X,X).

Returns:

(N, M) - Operator-valued kernels: (N, Dy, M, Dy)

Return type:

Scalar kernels

class dymad.modules.kernel.KernelScRBF(in_dim, lengthscale_init=None, dtype=None)¶

Bases: KernelScalarValued

Scalar RBF: k(x,z) = exp(-0.5 * ||x - z||^2 / ell^2) Learnable positive lengthscale.

property ell¶

forward(X, Z=None)¶

Compute kernel between X (N,d) and Z (M,d).

If Z is None, compute K(X,X).

Returns:

(N, M) - Operator-valued kernels: (N, Dy, M, Dy)

Return type:

Scalar kernels

set_reference_data(Xref)¶

Prepare data-dependent structures from Xref (N,d). Must be differentiable if kernel params are learnable.

By default the kernel is data-independent and does nothing.

Return type:: None

class dymad.modules.kernel.KernelScalarValued(in_dim, dtype=None)¶

Bases: KernelAbstract, ABC

property is_operator_valued: bool¶: True for operator-valued kernels; False for scalar kernels.

dymad.modules.kernel.inv_softplus(y, dtype)¶

Inverse of softplus, for initialization.

Return type:: Tensor

dymad.modules.kernel.scaled_cdist(X, Z, scale, p)¶

Pairwise distance ||X/scale - Z/scale||^p with broadcasting-friendly scaling.

Parameters:

X (torch.Tensor) – (N,d)
Z (torch.Tensor) – (M,d)
scale (float or torch.Tensor) – (d,) or scalar, positive
p (float) – order of the norm

Return type:

Tensor