Einstein Summation Made Easy: Using einsum for Efficient Tensor Manipulations in PyTorch

What is einsum?

• In linear algebra, Einstein summation notation is a concise way to represent sums over particular indices of tensors.
• PyTorch's torch.einsum function leverages this notation to perform efficient and expressive tensor operations.

How does einsum work?

1. Equation String: You provide a string defining the operation you want to perform.
2. Input Tensors: You pass these tensors as arguments following the equation string.
   • The order of tensors should match the order of their corresponding characters in the equation string.
3. Implicit Summation: einsum automatically sums over dimensions specified by repeated indices (appearing in both input and output operands).

Example:

import torch

# Sample tensors (batch size 2, feature dimensions 3 and 4)
a = torch.randn(2, 3)
b = torch.randn(2, 4)

# Matrix multiplication using einsum
c = torch.einsum("ab,bc->ac", a, b)  # Output shape: (2, 4)

# Equivalent matrix multiplication using @ operator
d = a @ b

print(c.shape, d.shape)  # Both will print (2, 4)

Benefits of einsum:

• Conciseness: It often expresses complex operations more compactly than traditional methods.
• Readability: The equation string can be easier to understand than nested loops or function calls.
• Flexibility: It supports various operations beyond matrix multiplication, such as element-wise operations, contractions, and more.
• Maintainability: Code using einsum can be easier to adapt to changes in tensor shapes or operations.

• When you need a concise and readable way to express complex tensor operations.
• When you're working with tensors of different shapes or want to perform custom contractions.
• When you want code that is adaptable to changes in tensor structure.

Additional Considerations:

• einsum is not necessarily faster than built-in operations like @ for simple matrix multiplication.
• It has a learning curve, but the benefits in readability and flexibility can outweigh the initial effort.

In summary, einsum is a powerful tool in PyTorch for performing efficient and expressive tensor operations. By understanding its syntax and how it leverages Einstein summation notation, you can write cleaner, more maintainable deep learning code.

Element-wise Product (Hadamard Product):

import torch

x = torch.randn(2, 3)
y = torch.randn(2, 3)

# Element-wise product using einsum
z = torch.einsum("ab,ab->ab", x, y)

# Equivalent element-wise product using * operator
w = x * y

print(z.shape, w.shape)  # Both will print (2, 3)

Batch Matrix Multiplication:

import torch

# Sample batch of matrices (batch size 3, matrices of size 4x5)
a = torch.randn(3, 4, 5)
b = torch.randn(3, 5, 6)

# Batch matrix multiplication using einsum
c = torch.einsum("bij,bjk->bik", a, b)  # Output shape: (3, 4, 6)

# Equivalent batch matrix multiplication using nested loops (less efficient)
d = []
for i in range(3):
  d.append(torch.matmul(a[i], b[i]))
d = torch.stack(d)

print(c.shape, d.shape)  # Both will print (3, 4, 6)

Vector Dot Product:

import torch

v1 = torch.randn(5)
v2 = torch.randn(5)

# Dot product using einsum
result = torch.einsum("i,i->", v1, v2)  # Output shape: () (scalar)

# Equivalent dot product using torch.dot
dot_product = torch.dot(v1, v2)

print(result.shape, dot_product.shape)  # Both will print ()

Transpose:

import torch

matrix = torch.randn(3, 4)

# Transpose using einsum
transposed = torch.einsum("ij->ji", matrix)  # Output shape: (4, 3)

# Equivalent transpose using .t() method
transposed_t = matrix.t()

print(transposed.shape, transposed_t.shape)  # Both will print (4, 3)

Summing Over Specific Dimensions:

import torch

data = torch.randn(2, 3, 4)

# Sum over dimension 1 (axis=1) using einsum
summed = torch.einsum("ijk->jk", data)  # Output shape: (2, 4)

# Equivalent sum using torch.sum with keepdim=False
reduced = torch.sum(data, dim=1, keepdim=False)

print(summed.shape, reduced.shape)  # Both will print (2, 4)

These examples showcase how einsum can be applied to various tensor operations, from basic arithmetic to more advanced manipulations. By understanding its syntax and capabilities, you can leverage it to write concise and expressive deep learning code in PyTorch.

Built-in PyTorch Operations:

• Matrix Multiplication: Use @ operator or torch.matmul(a, b).
• Element-wise Operations: Use element-wise arithmetic operators (+, -, *, /) or comparison operators (==, <, >).
• Vector Dot Product: Use torch.dot(v1, v2).
• Transpose: Use .t() method on the tensor (e.g., matrix.t()).
• Summation: Use torch.sum(tensor, dim, keepdim=False) for summation over specified dimensions.

Nested Loops (Less Efficient):

For custom operations involving loops over tensor elements, you can use nested loops. However, this approach can be less efficient and more error-prone compared to vectorized operations offered by einsum or built-in functions.

NumPy Conversion (if applicable):

If you're comfortable with NumPy and your tensors are small, you can convert them to NumPy arrays temporarily, perform the operation using np.einsum, and then convert back to PyTorch tensors. This might not be optimal for large tensors due to memory overhead.

Choosing the Right Method:

• For simple operations like matrix multiplication or element-wise product, built-in PyTorch operations are often the most efficient and readable choice.
• For more complex operations or when you need more control over summation, einsum can be a powerful and concise alternative.
• Nested loops should be a last resort due to potential performance and maintainability issues.
• NumPy conversion can be a workaround for small tensors but generally not recommended for large PyTorch tensors.

• Readability and Maintainability: Balance conciseness with clarity in your code.
• Performance: Consider the size and complexity of your tensors when choosing a method.
• Your Familiarity: Use the approach you're most comfortable with, but be open to learning new techniques.

By understanding these alternate methods and their trade-offs, you can make informed decisions when working with tensor operations in PyTorch.