Principle:LaurentMazare Tch rs Basic Tensor Operations

Knowledge Sources	LaurentMazare_Tch_rs
Domains	Numerical Computing, Deep Learning
Last Updated	2026-02-08 00:00 GMT

Overview

Fundamental tensor operations form the computational substrate upon which all numerical deep learning computations are expressed and executed.

Description

A tensor is a multi-dimensional array that generalizes scalars (rank 0), vectors (rank 1), and matrices (rank 2) to arbitrary dimensions. Tensor operations constitute the core primitives of any numerical computing framework and include:

• Creation: Constructing tensors from raw data, constant fills (zeros, ones, random), or by specifying shape and data type (dtype). Tensors carry metadata about their shape (dimensions), dtype (element type such as float32, int64), and device (CPU or GPU).

• Arithmetic: Element-wise operations (addition, subtraction, multiplication, division), matrix multiplication, broadcasting (automatic shape expansion for compatible dimensions), and reduction operations (sum, mean, max along axes).

• Device Transfer: Moving tensors between CPU and GPU memory. This is essential for leveraging hardware accelerators. Transfers are explicit operations that copy data across memory spaces.

• Automatic Differentiation (Autograd): Tensors can track computation history to enable reverse-mode automatic differentiation. When a tensor is created with gradient tracking enabled, all subsequent operations build a dynamic computation graph. Calling backward propagation on a scalar loss computes gradients for all tracked parameters.

Usage

These operations are applied in virtually every deep learning workflow: constructing input data, performing forward passes through neural networks, computing losses, and obtaining gradients for optimization. Understanding tensor semantics is prerequisite to building any model.

Theoretical Basis

Tensor Algebra Foundations:

A tensor $T$ of rank $n$ with shape $(d_1,d_2,\dots ,d_n)$ contains $\prod _{i=1}^n{d}_i$ elements.

Broadcasting Rules: Two tensors are broadcast-compatible if, for each trailing dimension, either:

1. The dimensions are equal, or
2. One of them is 1.

The resulting shape takes the maximum along each dimension.

Automatic Differentiation:

Given a computation graph producing scalar output $L$, reverse-mode autodiff computes:

$\frac{\partial L}{\partial x_i}$

for all input parameters $x_i$ by applying the chain rule backward through the graph:

$\frac{\partial L}{\partial x_i}=\sum _j\frac{\partial L}{\partial y_j}\cdot \frac{\partial y_j}{\partial x_i}$

where $y_j$ are intermediate variables that depend on $x_i$.

Device Transfer:

Tensor data resides in a specific memory space. Transfer between CPU and GPU involves:

1. Allocating memory on the target device
2. Copying element data across the bus (e.g., PCIe)
3. Releasing the source tensor (if moved, not copied)

Related Pages

• Implementation:LaurentMazare_Tch_rs_Basics_Example