Digit Recognition using Tensor Train (TT) Neural Networks

This notebook demonstrates how to use Tensor Train (TT) decomposition in neural network layers for image classification on the MNIST dataset.

Why TT Layers?

Traditional fully-connected layers have weight matrices of size input_dim × output_dim, which can become prohibitively large. TT layers decompose these weight matrices into a sequence of smaller 3D tensors (cores), dramatically reducing the number of parameters while maintaining expressiveness. This leads to:

• Fewer parameters → reduced memory footprint and faster inference

• Built-in regularization → less overfitting on small datasets

• Structured compression → the TT-rank controls the model capacity

Imports

import torch as tn
import torch.nn as nn
try:
    import torchtt as tntt
except:
    print('Installing torchTT...')
    %pip install git+https://github.com/ion-g-ion/torchTT
    import torchtt as tntt
from torch import optim
from torchvision import datasets
from torchvision.transforms import ToTensor
from torch.utils.data import DataLoader

device = tn.device('cuda' if tn.cuda.is_available() else 'cpu')

Load the MNIST Dataset

The MNIST dataset contains 70,000 grayscale images of handwritten digits (0-9), each of size 28×28 pixels:

• Training set: 60,000 images

• Test set: 10,000 images

We download the dataset and store it in the downloads folder. The ToTensor() transform converts PIL images to PyTorch tensors and normalizes pixel values to [0, 1].

train_data = datasets.MNIST(root = 'downloads', train = True, transform = ToTensor(), download = True)
test_data = datasets.MNIST(root = 'downloads', train = False, transform = ToTensor())

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Create DataLoaders

DataLoaders handle batching, shuffling, and parallel data loading:

• Training loader: batch size of 1000, shuffled each epoch for stochastic gradient descent

• Test loader: batch size of 100 for evaluation

Using num_workers=10 enables parallel data loading across multiple CPU processes.

dataloader_train = tn.utils.data.DataLoader(train_data, batch_size=1000, shuffle=True, num_workers=10)
dataloader_test = tn.utils.data.DataLoader(test_data, batch_size=100, shuffle=True, num_workers=10)

Define the TT Neural Network Architecture

The network uses two TT-Linear layers followed by a standard linear output layer:

Layer 1: LinearLayerTT

• Input shape: [1, 7, 4, 7, 4] — the 28×28 image reshaped as a 5D tensor (1×7×4×7×4 = 784)

• Output shape: [8, 10, 10, 10, 10] — produces 80,000 features in TT format

• TT-ranks: [1, 4, 2, 2, 2, 1] — controls the compression; lower ranks = fewer parameters

Layer 2: LinearLayerTT

• Input shape: [8, 10, 10, 10, 10]

• Output shape: [8, 3, 3, 3, 3] — produces 648 features (8×81)

• TT-ranks: [1, 2, 2, 2, 2, 1]

Output Layer

• Standard linear layer: 648 → 10 (one output per digit class)

• LogSoftmax activation for numerical stability with CrossEntropyLoss

ReLU activations are applied between the TT layers for non-linearity.

class BasicTT(nn.Module):
    def __init__(self):
        super().__init__()
        self.ttl1 = tntt.nn.LinearLayerTT([1,7,4,7,4], [8,10,10,10,10], [1,4,2,2,2,1])
        self.ttl2 = tntt.nn.LinearLayerTT([8,10,10,10,10], [8,3,3,3,3], [1,2,2,2,2,1])
        self.linear = nn.Linear(81*8, 10, dtype = tn.float32)
        self.logsoftmax = nn.LogSoftmax(1)

    def forward(self, x):
        x = self.ttl1(x)
        x = tn.relu(x)
        x = self.ttl2(x)
        x = tn.relu(x)
        x = x.view(-1,81*8)
        x = self.linear(x)
        return self.logsoftmax(x)

Instantiate Model, Loss, and Optimizer

• Model: Moved to GPU if available (CUDA), otherwise runs on CPU

• Loss function: CrossEntropyLoss combines LogSoftmax and NLLLoss — standard for multi-class classification

• Optimizer: Adam with learning rate 0.001 — adaptive learning rate optimizer that works well with default hyperparameters

model = BasicTT().to(device)
loss_function = nn.CrossEntropyLoss()
optimizer = optim.Adam(model.parameters(), lr = 0.001)

Training Loop

Train the model for 30 epochs. Each epoch:

1. Reshape input: Convert the flat 28×28 images to shape [-1, 1, 7, 4, 7, 4] to match the TT layer’s expected input format (batch dimension + 5D tensor structure)

2. Forward pass: Compute predictions through the TT network

3. Backward pass: Compute gradients via backpropagation

4. Update weights: Apply Adam optimizer step

Progress is printed after each batch showing epoch, iteration, and current loss value.

n_epochs = 30

for epoch in range(n_epochs):

    for i,(input,label) in enumerate(dataloader_train):

        input = tn.reshape(input.to(device),[-1,1,7,4,7,4])
        label = label.to(device)

        optimizer.zero_grad()
        output = model(input)
        loss = loss_function(output, label)
        loss.backward()
        optimizer.step()
        print('Epoch %d/%d iteration %d/%d loss %e'%(epoch+1,n_epochs,i+1,len(dataloader_train),loss))

/tmp/ipykernel_71331/1816520187.py:15: UserWarning: Converting a tensor with requires_grad=True to a scalar may lead to unexpected behavior.
Consider using tensor.detach() first. (Triggered internally at /pytorch/torch/csrc/autograd/generated/python_variable_methods.cpp:836.)
  print('Epoch %d/%d iteration %d/%d loss %e'%(epoch+1,n_epochs,i+1,len(dataloader_train),loss))

Epoch 1/30 iteration 1/60 loss 2.319157e+00
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Evaluate on Test Set

Compute the classification accuracy on the held-out test set:

1. Iterate through all test batches (no gradient computation needed)

2. For each sample, take the argmax of the output logits as the predicted class

3. Count correct predictions and compute accuracy = correct / total

A well-trained TT network should achieve ~97-98% accuracy on MNIST, comparable to standard fully-connected networks but with significantly fewer parameters.

n_correct = 0
n_total = 0
for (input,label) in dataloader_test:
    input = tn.reshape(input.to(device),[-1,1,7,4,7,4])

    output = model(input).cpu()

    n_correct += tn.sum(tn.max(output,1)[1] == label)

    n_total += input.shape[0]

print('Test accuracy ',n_correct/n_total)

Test accuracy  tensor(0.9735)