Linear Algebra

A Module based on Linear Algebra.

Implementation Notes

Tensor operations are unoptimized. I recommend PyTorch or TorchLib instead for a general Tensor Library.

Theoretical Motivation

Given the standard ring of $(\R, +, *)$ we can represent all potential operations as a composition of binary operators $f(x, y) = x * y$ or $f(x, y) = x + y$ where $x, y \in R$ . Given a vector space of $\R^n$ we can represent a vector as $\textbf{x} = \{(x_0, ..., x_n): x_i \in \R\}$ , where $f(\textbf{x}, \textbf{y})$ is some combination of $f(x_i, y_j)$ . Given a $p$ collection of vectors of size $n$ We define a matrix $\textbf{X} = (\textbf{x}_1, ..., \textbf{x}_p) \in \R^{n \times p}$ . We describe the structure of a Tensor-space $\mathfrak{T}$ of order $m$ on $\R$ as

\Large \mathfrak{T_\R^m} =\normalsize\prod_{i=1}^m \R^k,\enspace k \in \N^+

Given the usual ring $(\R, +, *)$ we can extend many theorems of linear algebra by allowing the following rule.

Given the tensors $\textbf{T}_1 = (\textbf{X}_1, ..., \textbf{X}_k)$ , $\textbf{T}_2 = (\textbf{Y}_1, ..., \textbf{Y}_p)$ , and $\textbf{T}_3 = (\textbf{Z}_1, ..., \textbf{Z}_q)$ a necessary (but not sufficient) condition for the operation $f(\textbf{T}_1, \textbf{T}_2) = \textbf{T}_3$ to be well defined is for $0 < i \leq max(p, k), \enspace dim(X_i) = dim(Y_i)$ , $dim(X_i) = 1$ , or $dim(Y_i) = 1$ and $dim(Z_i) = max(p, k)$ . Where WLOG $dim(X_i) = 1$ if $k < p$ .

Documentation

# pysiclib.linalg.Tensor

class Tensor:
    @overload
    def __init__(self, numpy_array: numpy.ndarray[numpy.float64]) -> None: ...
    @overload
    def __init__(self, input_data: List[float], input_shape: List[int] = ..., input_stride: List[int] = ..., offset: int = ...) -> None: ...
    @overload
    def __init__(self, other_view: Tensor) -> None: ...
    def binary_element_wise_op(self, arg0: Tensor, arg1: Callable[[float,float],float]) -> Tensor: ...
    def deep_copy(self) -> Tensor: ...
    def fold_op(self, arg0: Callable[[float,float],float], arg1: float, arg2: int, arg3: bool) -> Tensor: ...
    def get_buffer(self) -> List[float]: ...
    def get_offset(self) -> int: ...
    def get_shape(self) -> List[int]: ...
    def get_stride(self) -> List[int]: ...
    def matmul(self, arg0: Tensor) -> Tensor: ...
    def slice_view(self, arg0: List[int]) -> Tensor: ...
    def squeeze(self, target_dim: int = ...) -> Tensor: ...
    def to_numpy(self) -> numpy.ndarray[numpy.float64]: ...
    def transpose(self, dim_1: int = ..., dim_2: int = ...) -> Tensor: ...
    def unitary_op(self, arg0: Callable[[float],float]) -> Tensor: ...
    def unsqueeze(self, arg0: int) -> Tensor: ...