chebyshev series

src/beignet/polynomial/__init__.py

@@ -0,0 +1,3 @@
+from ._chebyshev import Chebyshev
+
+__all__ = ["Chebyshev"]

src/beignet/polynomial/_chebyshev.py

@@ -0,0 +1,243 @@
+import functools
+from typing import Callable
+
+import torch
+from torch import Tensor
+
+
+def _dct_ii_fft(input: Tensor, dim: int = -1):
+    input = input.transpose(dim, -1)
+
+    # see https://dsp.stackexchange.com/questions/2807/fast-cosine-transform-via-fft
+    N = input.shape[-1]
+
+    y = torch.zeros(*input.shape[:-1], 2 * N, device=input.device, dtype=input.dtype)
+    y[..., :N] = input
+    y[..., N:] = input.flip((-1,))
+
+    output = torch.fft.fft(y)[..., :N]
+
+    k = torch.arange(N, device=input.device, dtype=input.dtype)
+    output *= torch.exp(-1j * torch.pi * k / (2 * N))
+
+    output = output.transpose(dim, -1)
+
+    return output.real
+
+
+def _idct_ii_fft(input: Tensor, dim: int = -1):
+    input = input.transpose(dim, -1)
+
+    N = input.shape[-1]
+    k = torch.arange(N, dtype=input.dtype, device=input.device)
+
+    # Makhoul 9a
+    yk_half = torch.exp(1j * torch.pi * k / (2 * N)) * input
+
+    # Makhoul 12,13
+    yk = torch.cat(
+        [
+            yk_half,
+            torch.zeros(*input.shape[:-1], 1, dtype=input.dtype, device=input.device),
+            yk_half[..., 1:].conj().flip((-1,)),
+        ],
+        dim=-1,
+    )
+
+    yn = torch.fft.ifft(yk)
+
+    output = yn[..., :N].real
+
+    return output.transpose(dim, -1)
+
+
+@functools.lru_cache(maxsize=128)
+def chebyshev_t_roots(N: int, device=None, dtype=None):
+    k = torch.arange(N, device=device, dtype=dtype)
+    xk = torch.cos(torch.pi * (2 * k + 1) / (2 * N))
+    return xk
+
+
+def chebyshev_t_values_to_coefficients(input: Tensor, dim: int = -1):
+    input = input.transpose(dim, -1)
+
+    N = input.shape[-1]
+    j = torch.arange(0, N, dtype=input.dtype, device=input.device)
+    delta_j0 = (j == 0).to(input.dtype)
+    c = ((2 - delta_j0) / (2 * N)) * _dct_ii_fft(input)
+
+    c = c.transpose(dim, -1)
+    return c
+
+
+def chebyshev_t_coefficients_to_values(input: Tensor):
+    N = input.shape[-1]
+    j = torch.arange(0, N, dtype=input.dtype, device=input.device)
+    delta_j0 = (j == 0).to(input.dtype)
+    return _idct_ii_fft(input * (2 * N / (2 - delta_j0)))
+
+
+def chebyshev_t_clenshaw_eval(x: Tensor, a: Tensor):
+    # x: shape (*,)
+    # a: shape (**,N)
+    # output: shape (*, **)
+
+    N = a.shape[-1]
+
+    x = x[..., None]
+
+    bk_plus_one = torch.zeros_like(x)
+    bk_plus_two = torch.zeros_like(x)
+
+    for k in range(N - 1, 0, -1):
+        bk = a[..., k] + 2 * x * bk_plus_one - bk_plus_two
+        bk_plus_two = bk_plus_one
+        bk_plus_one = bk
+
+    # one more iteration to get b0
+    bk = a[..., 0] + 2 * x * bk_plus_one - bk_plus_two
+
+    return (0.5 * (a[..., 0] + bk - bk_plus_two)).squeeze(-1)
+
+
+def from_chebyshev_domain(x, a, b):
+    """Map x [-1, 1] to y in [a, b]."""
+    return x * (b - a) / 2 + (b + a) / 2
+
+
+def to_chebyshev_domain(y, a, b):
+    """Map y in [a, b] to x in [-1,1]."""
+    return (1 / (b - a)) * (2 * y - b - a)
+
+
+def chebyshev_t_integral_operator(N: int, dtype=None, device=None):
+    a = torch.cat(
+        [
+            torch.tensor(
+                [
+                    0.25,
+                ],
+                dtype=dtype,
+                device=device,
+            ),
+            -0.5 * (1 / (torch.arange(2, N + 1, dtype=dtype, device=device) - 1)),
+        ]
+    )
+    b = torch.cat(
+        [
+            torch.tensor([1, 0.25], dtype=dtype, device=device),
+            0.5 * (1 / (torch.arange(2, N, dtype=dtype, device=device) + 1)),
+        ]
+    )
+    op = (torch.diagflat(a, 1) + torch.diagflat(b, -1))[:, :-1]
+    return op
+
+
+def chebyshev_t_cumulative_integral_coefficients(
+    coefficients: Tensor,
+    a: float = -1.0,
+    b: float = 1.0,
+    dim: int = -1,
+    complement: bool = False,
+):
+    """Compute the coefficients for the chebyshev series for the cummulative integral."""
+    device = coefficients.device
+    dtype = coefficients.dtype
+    coefficients = coefficients.transpose(dim, -1)
+    N = coefficients.shape[-1]
+    op = chebyshev_t_integral_operator(N, device=device, dtype=dtype)
+    c_int = torch.einsum("ij,...j->...i", op, coefficients)
+    c_int *= (b - a) / 2
+
+    if complement:
+        int_b = c_int.sum(dim=-1)
+        c_int = -1 * c_int
+        c_int[..., 0] += int_b
+    else:
+        int_a = (c_int * (-1) ** torch.arange(N + 1, device=device, dtype=dtype)).sum(
+            dim=-1
+        )
+        c_int[..., 0] -= int_a
+
+    c_int = c_int.transpose(dim, -1)
+    return c_int
+
+
+class Chebyshev:
+    """Multidimensional chebyshev series representation."""
+
+    def __init__(
+        self,
+        coefficients: torch.Tensor,
+        a: list[float] | None,
+        b: list[float] | None,
+    ):
+        d = coefficients.ndim
+
+        assert d >= 0
+
+        if a is None:
+            a = [-1] * d
+
+        if b is None:
+            b = [1] * d
+
+        self.coefficients = coefficients
+        self.d = d
+        self.a = a
+        self.b = b
+
+    @classmethod
+    def fit(
+        cls,
+        f: Callable,
+        d: int,
+        order: int | list[int],
+        a: list[float] | None = None,
+        b: list[float] | None = None,
+        device=None,
+        dtype=None,
+    ):
+        if isinstance(order, int):
+            order = [order] * d
+
+        if a is None:
+            a = [-1] * d
+
+        if b is None:
+            b = [1] * d
+
+        y = []
+        for i in range(d):
+            shape = [order[i] if j == i else 1 for j in range(d)]
+            xi = chebyshev_t_roots(order[i], device=device, dtype=dtype)
+            yi = from_chebyshev_domain(xi, a[i], b[i]).view(*shape)
+            y.append(yi)
+
+        values = f(*y)
+
+        c = values
+        for i in range(d):
+            c = chebyshev_t_values_to_coefficients(c, dim=i)
+
+        return cls(c, a, b)
+
+    def __call__(self, *args):
+        assert len(args) == self.d
+
+        out = self.coefficients
+        for i in range(self.d - 1, -1, -1):
+            xi = to_chebyshev_domain(args[i], self.a[i], self.b[i])
+            out = chebyshev_t_clenshaw_eval(xi, out)
+
+        return out
+
+    def cumulative_integral(self, dim: int = -1, complement: bool = False):
+        c_int = chebyshev_t_cumulative_integral_coefficients(
+            self.coefficients,
+            a=self.a[dim],
+            b=self.b[dim],
+            dim=dim,
+            complement=complement,
+        )
+        return Chebyshev(c_int, self.a, self.b)

tests/beignet/polynomial/test__chebyshev.py

@@ -0,0 +1,53 @@
+import torch
+from beignet.polynomial import Chebyshev
+
+
+def test_chebyshev_fit_1d():
+    a = 0.0
+    b = 2.0
+
+    def f(x):
+        return torch.exp(-x + torch.cos(10 * x))
+
+    interp = Chebyshev.fit(f, d=1, a=[a], b=[b], order=200, dtype=torch.float64)
+
+    x = torch.linspace(a, b, 1001, dtype=torch.float64)
+
+    err = f(x) - interp(x)
+
+    assert (err.abs() < 1e-10).all()
+
+
+def test_chebyshev_fit_2d():
+    def f(x, y):
+        return torch.exp(-x * y + torch.cos(10 * x / (y.pow(2) + 1))) * torch.exp(
+            -y.pow(2) + 1.0
+        )
+
+    interp = Chebyshev.fit(f, d=2, order=[201, 101], dtype=torch.float64)
+
+    x = torch.linspace(-1.0, 1.0, 1001, dtype=torch.float64)
+    y = torch.linspace(-1.0, 1.0, 501, dtype=torch.float64)
+
+    err = f(x[:, None], y[None, :]) - interp(x, y)
+
+    assert (err.abs() < 1e-10).all()
+
+
+def test_chebyshev_cumulative_integral_2d():
+    def f(x, y):
+        return torch.sin(2 * x + 1) * y
+
+    def int_f(x, y):
+        return torch.sin(x) * torch.sin(x + 1) * y
+
+    interp = Chebyshev.fit(f, d=2, order=[201, 101], dtype=torch.float64)
+
+    int_interp = interp.cumulative_integral(dim=0)
+
+    x = torch.linspace(-1.0, 1.0, 1001, dtype=torch.float64)
+    y = torch.linspace(-1.0, 1.0, 501, dtype=torch.float64)
+
+    err = int_f(x[:, None], y[None, :]) - int_interp(x, y)
+
+    assert (err.abs() < 1e-10).all()