TinyTorch/modules/01_tensor/tensor_dev.ipynb

Module 01: Tensor Foundation - Building Blocks of ML

Welcome to Module 01! You're about to build the foundational Tensor class that powers all machine learning operations.

🔗 Prerequisites & Progress

You've Built: Nothing - this is our foundation! You'll Build: A complete Tensor class with arithmetic, matrix operations, and shape manipulation You'll Enable: Foundation for activations, layers, and all future neural network components

Connection Map:

NumPy Arrays → Tensor → Activations (Module 02)
(raw data)   (ML ops)  (intelligence)

Learning Objectives

By the end of this module, you will:

1. Implement a complete Tensor class with fundamental operations
2. Understand tensors as the universal data structure in ML
3. Test tensor operations with immediate validation
4. Prepare for gradient computation in Module 05

Let's get started!

📦 Where This Code Lives in the Final Package

Learning Side: You work in modules/01_tensor/tensor_dev.py Building Side: Code exports to tinytorch.core.tensor

# Final package structure:
# Future modules will import and extend this Tensor

Why this matters:

• Learning: Complete tensor system in one focused module for deep understanding
• Production: Proper organization like PyTorch's torch.Tensor with all core operations together
• Consistency: All tensor operations and data manipulation in core.tensor
• Integration: Foundation that every other module will build upon

In [ ]:

#| default_exp core.tensor
#| export

import numpy as np

1. Introduction: What is a Tensor?

A tensor is a multi-dimensional array that serves as the fundamental data structure in machine learning. Think of it as a universal container that can hold data in different dimensions:

Tensor Dimensions:
┌─────────────┐
│ 0D: Scalar  │  5.0          (just a number)
│ 1D: Vector  │  [1, 2, 3]    (list of numbers)
│ 2D: Matrix  │  [[1, 2]      (grid of numbers)
│             │   [3, 4]]
│ 3D: Cube    │  [[[...       (stack of matrices)
└─────────────┘

In machine learning, tensors flow through operations like water through pipes:

Neural Network Data Flow:
Input Tensor → Layer 1 → Activation → Layer 2 → ... → Output Tensor
   [batch,     [batch,     [batch,     [batch,          [batch,
    features]   hidden]     hidden]     hidden2]         classes]

Every neural network, from simple linear regression to modern transformers, processes tensors. Understanding tensors means understanding the foundation of all ML computations.

Why Tensors Matter in ML Systems

In production ML systems, tensors carry more than just data - they carry the computational graph, memory layout information, and execution context:

Real ML Pipeline:
Raw Data → Preprocessing → Tensor Creation → Model Forward Pass → Loss Computation
   ↓           ↓              ↓               ↓                    ↓
 Files     NumPy Arrays    Tensors        GPU Tensors         Scalar Loss

Key Insight: Tensors bridge the gap between mathematical concepts and efficient computation on modern hardware.

2. Foundations: Mathematical Background

Core Operations We'll Implement

Our Tensor class will support all fundamental operations that neural networks need:

Operation Types:
┌─────────────────┬─────────────────┬─────────────────┐
│ Element-wise    │ Matrix Ops      │ Shape Ops       │
├─────────────────┼─────────────────┼─────────────────┤
│ + Addition      │ @ Matrix Mult   │ .reshape()      │
│ - Subtraction   │ .transpose()    │ .sum()          │
│ * Multiplication│                 │ .mean()         │
│ / Division      │                 │ .max()          │
└─────────────────┴─────────────────┴─────────────────┘

Broadcasting: Making Tensors Work Together

Broadcasting automatically aligns tensors of different shapes for operations:

Broadcasting Examples:
┌─────────────────────────────────────────────────────────┐
│ Scalar + Vector:                                        │
│    5    + [1, 2, 3] → [5, 5, 5] + [1, 2, 3] = [6, 7, 8]│
│                                                         │
│ Matrix + Vector (row-wise):                             │
│ [[1, 2]]   [10]   [[1, 2]]   [[10, 10]]   [[11, 12]]  │
│ [[3, 4]] + [10] = [[3, 4]] + [[10, 10]] = [[13, 14]]  │
└─────────────────────────────────────────────────────────┘

Memory Layout: NumPy uses row-major (C-style) storage where elements are stored row by row in memory for cache efficiency:

Memory Layout (2×3 matrix):
Matrix:     Memory:
[[1, 2, 3]  [1][2][3][4][5][6]
 [4, 5, 6]]  ↑  Row 1   ↑  Row 2

Cache Behavior:
Sequential Access: Fast (uses cache lines efficiently)
  Row access: [1][2][3] → cache hit, hit, hit
Random Access: Slow (cache misses)
  Column access: [1][4] → cache hit, miss

This memory layout affects performance in real ML workloads - algorithms that access data sequentially run faster than those that access randomly.

3. Implementation: Building Tensor Foundation

Let's build our Tensor class step by step, testing each component as we go.

Key Design Decision: We'll include gradient-related attributes from the start, but they'll remain dormant until Module 05. This ensures a consistent interface throughout the course while keeping the cognitive load manageable.

Tensor Class Architecture

Tensor Class Structure:
┌─────────────────────────────────┐
│ Core Attributes:                │
│ • data: np.array (the numbers)  │
│ • shape: tuple (dimensions)     │
│ • size: int (total elements)    │
│ • dtype: type (float32, int64)  │
├─────────────────────────────────┤
│ Gradient Attributes (dormant):  │
│ • requires_grad: bool          │
│ • grad: None (until Module 05)  │
├─────────────────────────────────┤
│ Operations:                     │
│ • __add__, __sub__, __mul__     │
│ • matmul(), reshape()           │
│ • sum(), mean(), max()          │
│ • __repr__(), __str__()         │
└─────────────────────────────────┘

The beauty of this design: all methods are defined inside the class from day one. No monkey-patching, no dynamic attribute addition. Clean, consistent, debugger-friendly.

Tensor Creation and Initialization

Before we implement operations, let's understand how tensors store data and manage their attributes. This initialization is the foundation that everything else builds upon.

Tensor Initialization Process:
Input Data → Validation → NumPy Array → Tensor Wrapper → Ready for Operations
   [1,2,3] →    types   →  np.array   →    shape=(3,)  →     + - * / @ ...
     ↓             ↓          ↓             ↓
  List/Array    Type Check   Memory      Attributes Set
               (optional)    Allocation

Memory Allocation Example:
Input: [[1, 2, 3], [4, 5, 6]]
         ↓
NumPy allocates: [1][2][3][4][5][6] in contiguous memory
         ↓
Tensor wraps with: shape=(2,3), size=6, dtype=int64

Key Design Principle: Our Tensor is a wrapper around NumPy arrays that adds ML-specific functionality. We leverage NumPy's battle-tested memory management and computation kernels while adding the gradient tracking and operation chaining needed for deep learning.

Why This Approach?

• Performance: NumPy's C implementations are highly optimized
• Compatibility: Easy integration with scientific Python ecosystem
• Memory Efficiency: No unnecessary data copying
• Future-Proof: Easy transition to GPU tensors in advanced modules

In [ ]:

#| export
class Tensor:
    """Educational tensor that grows with student knowledge.

    This class starts simple but includes dormant features for future modules:
    - requires_grad: Will be used for automatic differentiation (Module 05)
    - grad: Will store computed gradients (Module 05)
    - backward(): Will compute gradients (Module 05)

    For now, focus on: data, shape, and basic operations.
    """

    def __init__(self, data, requires_grad=False):
        """
        Create a new tensor from data.

        TODO: Initialize tensor attributes

        APPROACH:
        1. Convert data to NumPy array - handles lists, scalars, etc.
        2. Store shape and size for quick access
        3. Set up gradient tracking (dormant until Module 05)

        EXAMPLE:
        >>> tensor = Tensor([1, 2, 3])
        >>> print(tensor.data)
        [1 2 3]
        >>> print(tensor.shape)
        (3,)

        HINT: np.array() handles type conversion automatically
        """
        ### BEGIN SOLUTION
        # Core tensor data - always present
        self.data = np.array(data, dtype=np.float32)  # Consistent float32 for ML
        self.shape = self.data.shape
        self.size = self.data.size
        self.dtype = self.data.dtype

        # Gradient features (dormant until Module 05)
        self.requires_grad = requires_grad
        self.grad = None
        ### END SOLUTION

    def __repr__(self):
        """String representation of tensor for debugging."""
        grad_info = f", requires_grad={self.requires_grad}" if self.requires_grad else ""
        return f"Tensor(data={self.data}, shape={self.shape}{grad_info})"

    def __str__(self):
        """Human-readable string representation."""
        return f"Tensor({self.data})"

    def numpy(self):
        """Return the underlying NumPy array."""
        return self.data

    # nbgrader={\"grade\": false, \"grade_id\": \"addition-impl\", \"solution\": true}
    def __add__(self, other):
        """
        Add two tensors element-wise with broadcasting support.

        TODO: Implement tensor addition with automatic broadcasting

        APPROACH:
        1. Handle both Tensor and scalar inputs
        2. Use NumPy's broadcasting for automatic shape alignment
        3. Return new Tensor with result (don't modify self)

        EXAMPLE:
        >>> a = Tensor([1, 2, 3])
        >>> b = Tensor([4, 5, 6])
        >>> result = a + b
        >>> print(result.data)
        [5. 7. 9.]

        BROADCASTING EXAMPLE:
        >>> matrix = Tensor([[1, 2], [3, 4]])  # Shape: (2, 2)
        >>> vector = Tensor([10, 20])          # Shape: (2,)
        >>> result = matrix + vector           # Broadcasting: (2,2) + (2,) → (2,2)
        >>> print(result.data)
        [[11. 22.]
         [13. 24.]]

        HINTS:
        - Use isinstance() to check if other is a Tensor
        - NumPy handles broadcasting automatically with +
        - Always return a new Tensor, don't modify self
        - Preserve gradient tracking for future modules
        """
        ### BEGIN SOLUTION
        if isinstance(other, Tensor):
            # Tensor + Tensor: let NumPy handle broadcasting
            return Tensor(self.data + other.data)
        else:
            # Tensor + scalar: NumPy broadcasts automatically
            return Tensor(self.data + other)
        ### END SOLUTION

    # nbgrader={"grade": false, "grade_id": "more-arithmetic", "solution": true}
    def __sub__(self, other):
        """
        Subtract two tensors element-wise.

        Common use: Centering data (x - mean), computing differences for loss functions.
        """
        ### BEGIN SOLUTION
        if isinstance(other, Tensor):
            return Tensor(self.data - other.data)
        else:
            return Tensor(self.data - other)
        ### END SOLUTION

    def __mul__(self, other):
        """
        Multiply two tensors element-wise (NOT matrix multiplication).

        Common use: Scaling features, applying masks, gating mechanisms in neural networks.
        Note: This is * operator, not @ (which will be matrix multiplication).
        """
        ### BEGIN SOLUTION
        if isinstance(other, Tensor):
            return Tensor(self.data * other.data)
        else:
            return Tensor(self.data * other)
        ### END SOLUTION

    def __truediv__(self, other):
        """
        Divide two tensors element-wise.

        Common use: Normalization (x / std), converting counts to probabilities.
        """
        ### BEGIN SOLUTION
        if isinstance(other, Tensor):
            return Tensor(self.data / other.data)
        else:
            return Tensor(self.data / other)
        ### END SOLUTION

    # nbgrader={"grade": false, "grade_id": "matmul-impl", "solution": true}
    def matmul(self, other):
        """
        Matrix multiplication of two tensors.

        TODO: Implement matrix multiplication using np.dot with proper validation

        APPROACH:
        1. Validate inputs are Tensors
        2. Check dimension compatibility (inner dimensions must match)
        3. Use np.dot for optimized computation
        4. Return new Tensor with result

        EXAMPLE:
        >>> a = Tensor([[1, 2], [3, 4]])  # 2×2
        >>> b = Tensor([[5, 6], [7, 8]])  # 2×2
        >>> result = a.matmul(b)          # 2×2 result
        >>> # Result: [[1×5+2×7, 1×6+2×8], [3×5+4×7, 3×6+4×8]] = [[19, 22], [43, 50]]

        SHAPE RULES:
        - (M, K) @ (K, N) → (M, N)  ✓ Valid
        - (M, K) @ (J, N) → Error   ✗ K ≠ J

        COMPLEXITY: O(M×N×K) for (M×K) @ (K×N) matrices

        HINTS:
        - np.dot handles the optimization for us
        - Check self.shape[-1] == other.shape[-2] for compatibility
        - Provide clear error messages for debugging
        """
        ### BEGIN SOLUTION
        if not isinstance(other, Tensor):
            raise TypeError(f"Expected Tensor for matrix multiplication, got {type(other)}")

        # Handle edge cases
        if self.shape == () or other.shape == ():
            # Scalar multiplication
            return Tensor(self.data * other.data)

        # For matrix multiplication, we need at least 1D tensors
        if len(self.shape) == 0 or len(other.shape) == 0:
            return Tensor(self.data * other.data)

        # Check dimension compatibility for matrix multiplication
        if len(self.shape) >= 2 and len(other.shape) >= 2:
            if self.shape[-1] != other.shape[-2]:
                raise ValueError(
                    f"Cannot perform matrix multiplication: {self.shape} @ {other.shape}. "
                    f"Inner dimensions must match: {self.shape[-1]} ≠ {other.shape[-2]}. "
                    f"💡 HINT: For (M,K) @ (K,N) → (M,N), the K dimensions must be equal."
                )
        elif len(self.shape) == 1 and len(other.shape) == 2:
            # Vector @ Matrix
            if self.shape[0] != other.shape[0]:
                raise ValueError(
                    f"Cannot multiply vector {self.shape} with matrix {other.shape}. "
                    f"Vector length {self.shape[0]} must match matrix rows {other.shape[0]}."
                )
        elif len(self.shape) == 2 and len(other.shape) == 1:
            # Matrix @ Vector
            if self.shape[1] != other.shape[0]:
                raise ValueError(
                    f"Cannot multiply matrix {self.shape} with vector {other.shape}. "
                    f"Matrix columns {self.shape[1]} must match vector length {other.shape[0]}."
                )

        # Perform optimized matrix multiplication
        # Use np.matmul (not np.dot) for proper batched matrix multiplication with 3D+ tensors
        result_data = np.matmul(self.data, other.data)
        return Tensor(result_data)
        ### END SOLUTION

    # nbgrader={"grade": false, "grade_id": "shape-ops", "solution": true}
    def reshape(self, *shape):
        """
        Reshape tensor to new dimensions.

        TODO: Implement tensor reshaping with validation

        APPROACH:
        1. Handle different calling conventions: reshape(2, 3) vs reshape((2, 3))
        2. Validate total elements remain the same
        3. Use NumPy's reshape for the actual operation
        4. Return new Tensor (keep immutability)

        EXAMPLE:
        >>> tensor = Tensor([1, 2, 3, 4, 5, 6])  # Shape: (6,)
        >>> reshaped = tensor.reshape(2, 3)      # Shape: (2, 3)
        >>> print(reshaped.data)
        [[1. 2. 3.]
         [4. 5. 6.]]

        COMMON USAGE:
        >>> # Flatten for MLP input
        >>> image = Tensor(np.random.rand(3, 32, 32))  # (channels, height, width)
        >>> flattened = image.reshape(-1)              # (3072,) - all pixels in vector
        >>>
        >>> # Prepare batch for convolution
        >>> batch = Tensor(np.random.rand(32, 784))    # (batch, features)
        >>> images = batch.reshape(32, 1, 28, 28)      # (batch, channels, height, width)

        HINTS:
        - Handle both reshape(2, 3) and reshape((2, 3)) calling styles
        - Check np.prod(new_shape) == self.size for validation
        - Use descriptive error messages for debugging
        """
        ### BEGIN SOLUTION
        # Handle both reshape(2, 3) and reshape((2, 3)) calling conventions
        if len(shape) == 1 and isinstance(shape[0], (tuple, list)):
            new_shape = tuple(shape[0])
        else:
            new_shape = shape

        # Handle -1 for automatic dimension inference (like NumPy)
        if -1 in new_shape:
            if new_shape.count(-1) > 1:
                raise ValueError("Can only specify one unknown dimension with -1")

            # Calculate the unknown dimension
            known_size = 1
            unknown_idx = new_shape.index(-1)
            for i, dim in enumerate(new_shape):
                if i != unknown_idx:
                    known_size *= dim

            unknown_dim = self.size // known_size
            new_shape = list(new_shape)
            new_shape[unknown_idx] = unknown_dim
            new_shape = tuple(new_shape)

        # Validate total elements remain the same
        if np.prod(new_shape) != self.size:
            raise ValueError(
                f"Cannot reshape tensor of size {self.size} to shape {new_shape}. "
                f"Total elements must match: {self.size} ≠ {np.prod(new_shape)}. "
                f"💡 HINT: Make sure new_shape dimensions multiply to {self.size}"
            )

        # Reshape the data (NumPy handles the memory layout efficiently)
        reshaped_data = np.reshape(self.data, new_shape)
        # Preserve gradient tracking from the original tensor (important for autograd!)
        result = Tensor(reshaped_data, requires_grad=self.requires_grad)
        return result
        ### END SOLUTION

    def transpose(self, dim0=None, dim1=None):
        """
        Transpose tensor dimensions.

        TODO: Implement tensor transposition

        APPROACH:
        1. Handle default case (transpose last two dimensions)
        2. Handle specific dimension swapping
        3. Use NumPy's transpose with proper axis specification
        4. Return new Tensor

        EXAMPLE:
        >>> matrix = Tensor([[1, 2, 3], [4, 5, 6]])  # (2, 3)
        >>> transposed = matrix.transpose()          # (3, 2)
        >>> print(transposed.data)
        [[1. 4.]
         [2. 5.]
         [3. 6.]]

        NEURAL NETWORK USAGE:
        >>> # Weight matrix transpose for backward pass
        >>> W = Tensor([[0.1, 0.2], [0.3, 0.4], [0.5, 0.6]])  # (3, 2)
        >>> W_T = W.transpose()  # (2, 3) - for gradient computation
        >>>
        >>> # Attention mechanism
        >>> Q = Tensor([[1, 2], [3, 4]])  # queries (2, 2)
        >>> K = Tensor([[5, 6], [7, 8]])  # keys (2, 2)
        >>> attention_scores = Q.matmul(K.transpose())  # Q @ K^T

        HINTS:
        - Default: transpose last two dimensions (most common case)
        - Use np.transpose() with axes parameter
        - Handle 1D tensors gracefully (transpose is identity)
        """
        ### BEGIN SOLUTION
        if dim0 is None and dim1 is None:
            # Default: transpose last two dimensions
            if len(self.shape) < 2:
                # For 1D tensors, transpose is identity operation
                return Tensor(self.data.copy())
            else:
                # Transpose last two dimensions (most common in ML)
                axes = list(range(len(self.shape)))
                axes[-2], axes[-1] = axes[-1], axes[-2]
                transposed_data = np.transpose(self.data, axes)
        else:
            # Specific dimensions to transpose
            if dim0 is None or dim1 is None:
                raise ValueError("Both dim0 and dim1 must be specified for specific dimension transpose")

            # Validate dimensions exist
            if dim0 >= len(self.shape) or dim1 >= len(self.shape) or dim0 < 0 or dim1 < 0:
                raise ValueError(
                    f"Dimension out of range for tensor with shape {self.shape}. "
                    f"Got dim0={dim0}, dim1={dim1}, but tensor has {len(self.shape)} dimensions."
                )

            # Create axes list and swap the specified dimensions
            axes = list(range(len(self.shape)))
            axes[dim0], axes[dim1] = axes[dim1], axes[dim0]
            transposed_data = np.transpose(self.data, axes)

        # Preserve requires_grad for gradient tracking (Module 05 will add _grad_fn)
        result = Tensor(transposed_data, requires_grad=self.requires_grad if hasattr(self, 'requires_grad') else False)
        return result
        ### END SOLUTION

    # nbgrader={"grade": false, "grade_id": "reduction-ops", "solution": true}
    def sum(self, axis=None, keepdims=False):
        """
        Sum tensor along specified axis.

        TODO: Implement tensor sum with axis control

        APPROACH:
        1. Use NumPy's sum with axis parameter
        2. Handle axis=None (sum all elements) vs specific axis
        3. Support keepdims to maintain shape for broadcasting
        4. Return new Tensor with result

        EXAMPLE:
        >>> tensor = Tensor([[1, 2], [3, 4]])
        >>> total = tensor.sum()          # Sum all elements: 10
        >>> col_sum = tensor.sum(axis=0)  # Sum columns: [4, 6]
        >>> row_sum = tensor.sum(axis=1)  # Sum rows: [3, 7]

        NEURAL NETWORK USAGE:
        >>> # Batch loss computation
        >>> batch_losses = Tensor([0.1, 0.3, 0.2, 0.4])  # Individual losses
        >>> total_loss = batch_losses.sum()               # Total: 1.0
        >>> avg_loss = batch_losses.mean()                # Average: 0.25
        >>>
        >>> # Global average pooling
        >>> feature_maps = Tensor(np.random.rand(32, 256, 7, 7))  # (batch, channels, h, w)
        >>> global_features = feature_maps.sum(axis=(2, 3))       # (batch, channels)

        HINTS:
        - np.sum handles all the complexity for us
        - axis=None sums all elements (returns scalar)
        - axis=0 sums along first dimension, axis=1 along second, etc.
        - keepdims=True preserves dimensions for broadcasting
        """
        ### BEGIN SOLUTION
        result = np.sum(self.data, axis=axis, keepdims=keepdims)
        return Tensor(result)
        ### END SOLUTION

    def mean(self, axis=None, keepdims=False):
        """
        Compute mean of tensor along specified axis.

        Common usage: Batch normalization, loss averaging, global pooling.
        """
        ### BEGIN SOLUTION
        result = np.mean(self.data, axis=axis, keepdims=keepdims)
        return Tensor(result)
        ### END SOLUTION

    def max(self, axis=None, keepdims=False):
        """
        Find maximum values along specified axis.

        Common usage: Max pooling, finding best predictions, activation clipping.
        """
        ### BEGIN SOLUTION
        result = np.max(self.data, axis=axis, keepdims=keepdims)
        return Tensor(result)
        ### END SOLUTION

    # nbgrader={"grade": false, "grade_id": "gradient-placeholder", "solution": true}
    def backward(self):
        """
        Compute gradients (implemented in Module 05: Autograd).

        TODO: Placeholder implementation for gradient computation

        STUDENT NOTE:
        This method exists but does nothing until Module 05: Autograd.
        Don't worry about it for now - focus on the basic tensor operations.

        In Module 05, we'll implement:
        - Gradient computation via chain rule
        - Automatic differentiation
        - Backpropagation through operations
        - Computation graph construction

        FUTURE IMPLEMENTATION PREVIEW:
        ```python
        def backward(self, gradient=None):
            # Module 05 will implement:
            # 1. Set gradient for this tensor
            # 2. Propagate to parent operations
            # 3. Apply chain rule recursively
            # 4. Accumulate gradients properly
            pass
        ```

        CURRENT BEHAVIOR:
        >>> x = Tensor([1, 2, 3], requires_grad=True)
        >>> y = x * 2
        >>> y.sum().backward()  # Calls this method - does nothing
        >>> print(x.grad)      # Still None
        None
        """
        ### BEGIN SOLUTION
        # Placeholder - will be implemented in Module 05
        # For now, just ensure it doesn't crash when called
        # This allows students to experiment with gradient syntax
        # without getting confusing errors about missing methods
        pass
        ### END SOLUTION

🧪 Unit Test: Tensor Creation

This test validates our Tensor constructor works correctly with various data types and properly initializes all attributes.

What we're testing: Basic tensor creation and attribute setting Why it matters: Foundation for all other operations - if creation fails, nothing works Expected: Tensor wraps data correctly with proper attributes and consistent dtype

In [ ]:

def test_unit_tensor_creation():
    """🧪 Test Tensor creation with various data types."""
    print("🧪 Unit Test: Tensor Creation...")

    # Test scalar creation
    scalar = Tensor(5.0)
    assert scalar.data == 5.0
    assert scalar.shape == ()
    assert scalar.size == 1
    assert scalar.requires_grad == False
    assert scalar.grad is None
    assert scalar.dtype == np.float32

    # Test vector creation
    vector = Tensor([1, 2, 3])
    assert np.array_equal(vector.data, np.array([1, 2, 3], dtype=np.float32))
    assert vector.shape == (3,)
    assert vector.size == 3

    # Test matrix creation
    matrix = Tensor([[1, 2], [3, 4]])
    assert np.array_equal(matrix.data, np.array([[1, 2], [3, 4]], dtype=np.float32))
    assert matrix.shape == (2, 2)
    assert matrix.size == 4

    # Test gradient flag (dormant feature)
    grad_tensor = Tensor([1, 2], requires_grad=True)
    assert grad_tensor.requires_grad == True
    assert grad_tensor.grad is None  # Still None until Module 05

    print("✅ Tensor creation works correctly!")

if __name__ == "__main__":
    test_unit_tensor_creation()

Element-wise Arithmetic Operations

Element-wise operations are the workhorses of neural network computation. They apply the same operation to corresponding elements in tensors, often with broadcasting to handle different shapes elegantly.

Why Element-wise Operations Matter

In neural networks, element-wise operations appear everywhere:

• Activation functions: Apply ReLU, sigmoid to every element
• Batch normalization: Subtract mean, divide by std per element
• Loss computation: Compare predictions vs. targets element-wise
• Gradient updates: Add scaled gradients to parameters element-wise

Element-wise Addition: The Foundation

Addition is the simplest and most fundamental operation. Understanding it deeply helps with all others.

Element-wise Addition Visual:
[1, 2, 3] + [4, 5, 6] = [1+4, 2+5, 3+6] = [5, 7, 9]

Matrix Addition:
[[1, 2]]   [[5, 6]]   [[1+5, 2+6]]   [[6, 8]]
[[3, 4]] + [[7, 8]] = [[3+7, 4+8]] = [[10, 12]]

Broadcasting Addition (Matrix + Vector):
[[1, 2]]   [10]   [[1, 2]]   [[10, 10]]   [[11, 12]]
[[3, 4]] + [20] = [[3, 4]] + [[20, 20]] = [[23, 24]]
     ↑      ↑           ↑         ↑            ↑
  (2,2)   (2,1)      (2,2)    broadcast    result

Broadcasting Rules:
1. Start from rightmost dimension
2. Dimensions must be equal OR one must be 1 OR one must be missing
3. Missing dimensions are assumed to be 1

Key Insight: Broadcasting makes tensors of different shapes compatible by automatically expanding dimensions. This is crucial for batch processing where you often add a single bias vector to an entire batch of data.

Memory Efficiency: Broadcasting doesn't actually create expanded copies in memory - NumPy computes results on-the-fly, saving memory.

Subtraction, Multiplication, and Division

These operations follow the same pattern as addition, working element-wise with broadcasting support. Each serves specific purposes in neural networks:

Element-wise Operations in Neural Networks:

┌─────────────────┬─────────────────┬─────────────────┬─────────────────┐
│ Subtraction     │ Multiplication  │ Division        │ Use Cases       │
├─────────────────┼─────────────────┼─────────────────┼─────────────────┤
│ [6,8] - [1,2]   │ [2,3] * [4,5]   │ [8,9] / [2,3]   │ • Gradient      │
│ = [5,6]         │ = [8,15]        │ = [4.0, 3.0]    │   computation   │
│                 │                 │                 │ • Normalization │
│ Center data:    │ Gate values:    │ Scale features: │ • Loss functions│
│ x - mean        │ x * mask        │ x / std         │ • Attention     │
└─────────────────┴─────────────────┴─────────────────┴─────────────────┘

Broadcasting with Scalars (very common in ML):
[1, 2, 3] * 2     = [2, 4, 6]      (scale all values)
[1, 2, 3] - 1     = [0, 1, 2]      (shift all values)
[2, 4, 6] / 2     = [1, 2, 3]      (normalize all values)

Real ML Example - Batch Normalization:
batch_data = [[1, 2], [3, 4], [5, 6]]  # Shape: (3, 2)
mean = [3, 4]                           # Shape: (2,)
std = [2, 2]                            # Shape: (2,)

# Normalize: (x - mean) / std
normalized = (batch_data - mean) / std
# Broadcasting: (3,2) - (2,) = (3,2), then (3,2) / (2,) = (3,2)

Performance Note: Element-wise operations are highly optimized in NumPy and run efficiently on modern CPUs with vectorization (SIMD instructions).

🧪 Unit Test: Arithmetic Operations

This test validates our arithmetic operations work correctly with both tensor-tensor and tensor-scalar operations, including broadcasting behavior.

What we're testing: Addition, subtraction, multiplication, division with broadcasting Why it matters: Foundation for neural network forward passes, batch processing, normalization Expected: Operations work with both tensors and scalars, proper broadcasting alignment

In [ ]:

def test_unit_arithmetic_operations():
    """🧪 Test arithmetic operations with broadcasting."""
    print("🧪 Unit Test: Arithmetic Operations...")

    # Test tensor + tensor
    a = Tensor([1, 2, 3])
    b = Tensor([4, 5, 6])
    result = a + b
    assert np.array_equal(result.data, np.array([5, 7, 9], dtype=np.float32))

    # Test tensor + scalar (very common in ML)
    result = a + 10
    assert np.array_equal(result.data, np.array([11, 12, 13], dtype=np.float32))

    # Test broadcasting with different shapes (matrix + vector)
    matrix = Tensor([[1, 2], [3, 4]])
    vector = Tensor([10, 20])
    result = matrix + vector
    expected = np.array([[11, 22], [13, 24]], dtype=np.float32)
    assert np.array_equal(result.data, expected)

    # Test subtraction (data centering)
    result = b - a
    assert np.array_equal(result.data, np.array([3, 3, 3], dtype=np.float32))

    # Test multiplication (scaling)
    result = a * 2
    assert np.array_equal(result.data, np.array([2, 4, 6], dtype=np.float32))

    # Test division (normalization)
    result = b / 2
    assert np.array_equal(result.data, np.array([2.0, 2.5, 3.0], dtype=np.float32))

    # Test chaining operations (common in ML pipelines)
    normalized = (a - 2) / 2  # Center and scale
    expected = np.array([-0.5, 0.0, 0.5], dtype=np.float32)
    assert np.allclose(normalized.data, expected)

    print("✅ Arithmetic operations work correctly!")

if __name__ == "__main__":
    test_unit_arithmetic_operations()

Matrix Multiplication: The Heart of Neural Networks

Matrix multiplication is fundamentally different from element-wise multiplication. It's the operation that gives neural networks their power to transform and combine information across features.

Why Matrix Multiplication is Central to ML

Every neural network layer essentially performs matrix multiplication:

Linear Layer (the building block of neural networks):
Input Features × Weight Matrix = Output Features
    (N, D_in)   ×    (D_in, D_out)  =    (N, D_out)

Real Example - Image Classification:
Flattened Image × Hidden Weights = Hidden Features
  (32, 784)     ×    (784, 256)   =   (32, 256)
     ↑                   ↑              ↑
  32 images         784→256 transform  32 feature vectors

Matrix Multiplication Visualization

Matrix Multiplication Process:
    A (2×3)      B (3×2)         C (2×2)
   ┌       ┐    ┌     ┐       ┌         ┐
   │ 1 2 3 │    │ 7 8 │       │ 1×7+2×9+3×1 │   ┌      ┐
   │       │ ×  │ 9 1 │  =    │             │ = │ 28 13│
   │ 4 5 6 │    │ 1 2 │       │ 4×7+5×9+6×1 │   │ 79 37│
   └       ┘    └     ┘       └             ┘   └      ┘

Computation Breakdown:
C[0,0] = A[0,:] · B[:,0] = [1,2,3] · [7,9,1] = 1×7 + 2×9 + 3×1 = 28
C[0,1] = A[0,:] · B[:,1] = [1,2,3] · [8,1,2] = 1×8 + 2×1 + 3×2 = 13
C[1,0] = A[1,:] · B[:,0] = [4,5,6] · [7,9,1] = 4×7 + 5×9 + 6×1 = 79
C[1,1] = A[1,:] · B[:,1] = [4,5,6] · [8,1,2] = 4×8 + 5×1 + 6×2 = 37

Key Rule: Inner dimensions must match!
A(m,n) @ B(n,p) = C(m,p)
     ↑     ↑
   these must be equal

Computational Complexity and Performance

Computational Cost:
For C = A @ B where A is (M×K), B is (K×N):
- Multiplications: M × N × K
- Additions: M × N × (K-1) ≈ M × N × K
- Total FLOPs: ≈ 2 × M × N × K

Example: (1000×1000) @ (1000×1000)
- FLOPs: 2 × 1000³ = 2 billion operations
- On 1 GHz CPU: ~2 seconds if no optimization
- With optimized BLAS: ~0.1 seconds (20× speedup!)

Memory Access Pattern:
A: M×K (row-wise access)  ✓ Good cache locality
B: K×N (column-wise)      ✗ Poor cache locality
C: M×N (row-wise write)   ✓ Good cache locality

This is why optimized libraries like OpenBLAS, Intel MKL use:
- Blocking algorithms (process in cache-sized chunks)
- Vectorization (SIMD instructions)
- Parallelization (multiple cores)

Neural Network Context

Multi-layer Neural Network:
Input (batch=32, features=784)
  ↓ W1: (784, 256)
Hidden1 (batch=32, features=256)
  ↓ W2: (256, 128)
Hidden2 (batch=32, features=128)
  ↓ W3: (128, 10)
Output (batch=32, classes=10)

Each arrow represents a matrix multiplication:
- Forward pass: 3 matrix multiplications
- Backward pass: 3 more matrix multiplications (with transposes)
- Total: 6 matrix mults per forward+backward pass

For training batch: 32 × (784×256 + 256×128 + 128×10) FLOPs
= 32 × (200,704 + 32,768 + 1,280) = 32 × 234,752 = 7.5M FLOPs per batch

This is why GPU acceleration matters - modern GPUs can perform thousands of these operations in parallel!

🧪 Unit Test: Matrix Multiplication

This test validates matrix multiplication works correctly with proper shape checking and error handling.

What we're testing: Matrix multiplication with shape validation and edge cases Why it matters: Core operation in neural networks (linear layers, attention mechanisms) Expected: Correct results for valid shapes, clear error messages for invalid shapes

In [ ]:

def test_unit_matrix_multiplication():
    """🧪 Test matrix multiplication operations."""
    print("🧪 Unit Test: Matrix Multiplication...")

    # Test 2×2 matrix multiplication (basic case)
    a = Tensor([[1, 2], [3, 4]])  # 2×2
    b = Tensor([[5, 6], [7, 8]])  # 2×2
    result = a.matmul(b)
    # Expected: [[1×5+2×7, 1×6+2×8], [3×5+4×7, 3×6+4×8]] = [[19, 22], [43, 50]]
    expected = np.array([[19, 22], [43, 50]], dtype=np.float32)
    assert np.array_equal(result.data, expected)

    # Test rectangular matrices (common in neural networks)
    c = Tensor([[1, 2, 3], [4, 5, 6]])  # 2×3 (like batch_size=2, features=3)
    d = Tensor([[7, 8], [9, 10], [11, 12]])  # 3×2 (like features=3, outputs=2)
    result = c.matmul(d)
    # Expected: [[1×7+2×9+3×11, 1×8+2×10+3×12], [4×7+5×9+6×11, 4×8+5×10+6×12]]
    expected = np.array([[58, 64], [139, 154]], dtype=np.float32)
    assert np.array_equal(result.data, expected)

    # Test matrix-vector multiplication (common in forward pass)
    matrix = Tensor([[1, 2, 3], [4, 5, 6]])  # 2×3
    vector = Tensor([1, 2, 3])  # 3×1 (conceptually)
    result = matrix.matmul(vector)
    # Expected: [1×1+2×2+3×3, 4×1+5×2+6×3] = [14, 32]
    expected = np.array([14, 32], dtype=np.float32)
    assert np.array_equal(result.data, expected)

    # Test shape validation - should raise clear error
    try:
        incompatible_a = Tensor([[1, 2]])     # 1×2
        incompatible_b = Tensor([[1], [2], [3]])  # 3×1
        incompatible_a.matmul(incompatible_b)  # 1×2 @ 3×1 should fail (2 ≠ 3)
        assert False, "Should have raised ValueError for incompatible shapes"
    except ValueError as e:
        assert "Inner dimensions must match" in str(e)
        assert "2 ≠ 3" in str(e)  # Should show specific dimensions

    print("✅ Matrix multiplication works correctly!")

if __name__ == "__main__":
    test_unit_matrix_multiplication()

Shape Manipulation: Reshape and Transpose

Neural networks constantly change tensor shapes to match layer requirements. Understanding these operations is crucial for data flow through networks.

Why Shape Manipulation Matters

Real neural networks require constant shape changes:

CNN Data Flow Example:
Input Image: (32, 3, 224, 224)     # batch, channels, height, width
     ↓ Convolutional layers
Feature Maps: (32, 512, 7, 7)      # batch, features, spatial
     ↓ Global Average Pool
Pooled: (32, 512, 1, 1)            # batch, features, 1, 1
     ↓ Flatten for classifier
Flattened: (32, 512)               # batch, features
     ↓ Linear classifier
Output: (32, 1000)                 # batch, classes

Each ↓ involves reshape or view operations!

Reshape: Changing Interpretation of the Same Data

Reshaping (changing dimensions without changing data):
Original: [1, 2, 3, 4, 5, 6]  (shape: (6,))
         ↓ reshape(2, 3)
Result:  [[1, 2, 3],          (shape: (2, 3))
          [4, 5, 6]]

Memory Layout (unchanged):
Before: [1][2][3][4][5][6]
After:  [1][2][3][4][5][6]  ← Same memory, different interpretation

Key Insight: Reshape is O(1) operation - no data copying!
Just changes how we interpret the memory layout.

Common ML Reshapes:
┌─────────────────────┬─────────────────────┬─────────────────────┐
│ Flatten for MLP     │ Unflatten for CNN   │ Batch Dimension     │
├─────────────────────┼─────────────────────┼─────────────────────┤
│ (N,H,W,C) → (N,H×W×C) │ (N,D) → (N,H,W,C)   │ (H,W) → (1,H,W)     │
│ Images to vectors   │ Vectors to images   │ Add batch dimension │
└─────────────────────┴─────────────────────┴─────────────────────┘

Transpose: Swapping Dimensions

Transposing (swapping dimensions - data rearrangement):
Original: [[1, 2, 3],    (shape: (2, 3))
           [4, 5, 6]]
         ↓ transpose()
Result:  [[1, 4],        (shape: (3, 2))
          [2, 5],
          [3, 6]]

Memory Layout (rearranged):
Before: [1][2][3][4][5][6]
After:  [1][4][2][5][3][6]  ← Data actually moves in memory

Key Insight: Transpose involves data movement - more expensive than reshape.

Neural Network Usage:
┌─────────────────────┬─────────────────────┬─────────────────────┐
│ Weight Matrices     │ Attention Mechanism │ Gradient Computation│
├─────────────────────┼─────────────────────┼─────────────────────┤
│ Forward: X @ W      │ Q @ K^T attention   │ ∂L/∂W = X^T @ ∂L/∂Y│
│ Backward: X @ W^T   │ scores              │                     │
└─────────────────────┴─────────────────────┴─────────────────────┘

Performance Implications

Operation Performance (for 1000×1000 matrix):
┌─────────────────┬──────────────┬─────────────────┬─────────────────┐
│ Operation       │ Time         │ Memory Access   │ Cache Behavior  │
├─────────────────┼──────────────┼─────────────────┼─────────────────┤
│ reshape()       │ ~0.001 ms    │ No data copy    │ No cache impact │
│ transpose()     │ ~10 ms       │ Full data copy  │ Poor locality   │
│ view() (future) │ ~0.001 ms    │ No data copy    │ No cache impact │
└─────────────────┴──────────────┴─────────────────┴─────────────────┘

Why transpose() is slower:
- Must rearrange data in memory
- Poor cache locality (accessing columns)
- Can't be parallelized easily

This is why frameworks like PyTorch often use "lazy" transpose operations that defer the actual data movement until necessary.

🧪 Unit Test: Shape Manipulation

This test validates reshape and transpose operations work correctly with validation and edge cases.

What we're testing: Reshape and transpose operations with proper error handling Why it matters: Essential for data flow in neural networks, CNN/RNN architectures Expected: Correct shape changes, proper error handling for invalid operations

In [ ]:

def test_unit_shape_manipulation():
    """🧪 Test reshape and transpose operations."""
    print("🧪 Unit Test: Shape Manipulation...")

    # Test basic reshape (flatten → matrix)
    tensor = Tensor([1, 2, 3, 4, 5, 6])  # Shape: (6,)
    reshaped = tensor.reshape(2, 3)      # Shape: (2, 3)
    assert reshaped.shape == (2, 3)
    expected = np.array([[1, 2, 3], [4, 5, 6]], dtype=np.float32)
    assert np.array_equal(reshaped.data, expected)

    # Test reshape with tuple (alternative calling style)
    reshaped2 = tensor.reshape((3, 2))   # Shape: (3, 2)
    assert reshaped2.shape == (3, 2)
    expected2 = np.array([[1, 2], [3, 4], [5, 6]], dtype=np.float32)
    assert np.array_equal(reshaped2.data, expected2)

    # Test reshape with -1 (automatic dimension inference)
    auto_reshaped = tensor.reshape(2, -1)  # Should infer -1 as 3
    assert auto_reshaped.shape == (2, 3)

    # Test reshape validation - should raise error for incompatible sizes
    try:
        tensor.reshape(2, 2)  # 6 elements can't fit in 2×2=4
        assert False, "Should have raised ValueError"
    except ValueError as e:
        assert "Total elements must match" in str(e)
        assert "6 ≠ 4" in str(e)

    # Test matrix transpose (most common case)
    matrix = Tensor([[1, 2, 3], [4, 5, 6]])  # (2, 3)
    transposed = matrix.transpose()          # (3, 2)
    assert transposed.shape == (3, 2)
    expected = np.array([[1, 4], [2, 5], [3, 6]], dtype=np.float32)
    assert np.array_equal(transposed.data, expected)

    # Test 1D transpose (should be identity)
    vector = Tensor([1, 2, 3])
    vector_t = vector.transpose()
    assert np.array_equal(vector.data, vector_t.data)

    # Test specific dimension transpose
    tensor_3d = Tensor([[[1, 2], [3, 4]], [[5, 6], [7, 8]]])  # (2, 2, 2)
    swapped = tensor_3d.transpose(0, 2)  # Swap first and last dimensions
    assert swapped.shape == (2, 2, 2)  # Same shape but data rearranged

    # Test neural network reshape pattern (flatten for MLP)
    batch_images = Tensor(np.random.rand(2, 3, 4))  # (batch=2, height=3, width=4)
    flattened = batch_images.reshape(2, -1)  # (batch=2, features=12)
    assert flattened.shape == (2, 12)

    print("✅ Shape manipulation works correctly!")

if __name__ == "__main__":
    test_unit_shape_manipulation()

Reduction Operations: Aggregating Information

Reduction operations collapse dimensions by aggregating data, which is essential for computing statistics, losses, and preparing data for different layers.

Why Reductions are Crucial in ML

Reduction operations appear throughout neural networks:

Common ML Reduction Patterns:

┌─────────────────────┬─────────────────────┬─────────────────────┐
│ Loss Computation    │ Batch Normalization │ Global Pooling      │
├─────────────────────┼─────────────────────┼─────────────────────┤
│ Per-sample losses → │ Batch statistics →  │ Feature maps →      │
│ Single batch loss   │ Normalization       │ Single features     │
│                     │                     │                     │
│ losses.mean()       │ batch.mean(axis=0)  │ fmaps.mean(axis=(2,3))│
│ (N,) → scalar       │ (N,D) → (D,)        │ (N,C,H,W) → (N,C)   │
└─────────────────────┴─────────────────────┴─────────────────────┘

Real Examples:
• Cross-entropy loss: -log(predictions).mean()  [average over batch]
• Batch norm: (x - x.mean()) / x.std()          [normalize each feature]
• Global avg pool: features.mean(dim=(2,3))     [spatial → scalar per channel]

Understanding Axis Operations

Visual Axis Understanding:
Matrix:     [[1, 2, 3],      All reductions operate on this data
             [4, 5, 6]]      Shape: (2, 3)

        axis=0 (↓)
       ┌─────────┐
axis=1 │ 1  2  3 │ →  axis=1 reduces across columns (→)
   (→) │ 4  5  6 │ →  Result shape: (2,) [one value per row]
       └─────────┘
         ↓ ↓ ↓
      axis=0 reduces down rows (↓)
      Result shape: (3,) [one value per column]

Reduction Results:
├─ .sum() → 21                    (sum all: 1+2+3+4+5+6)
├─ .sum(axis=0) → [5, 7, 9]       (sum columns: [1+4, 2+5, 3+6])
├─ .sum(axis=1) → [6, 15]         (sum rows: [1+2+3, 4+5+6])
├─ .mean() → 3.5                  (average all: 21/6)
├─ .mean(axis=0) → [2.5, 3.5, 4.5] (average columns)
└─ .max() → 6                     (maximum element)

3D Tensor Example (batch, height, width):
data.shape = (2, 3, 4)  # 2 samples, 3×4 images
│
├─ .sum(axis=0) → (3, 4)    # Sum across batch dimension
├─ .sum(axis=1) → (2, 4)    # Sum across height dimension
├─ .sum(axis=2) → (2, 3)    # Sum across width dimension
└─ .sum(axis=(1,2)) → (2,)  # Sum across both spatial dims (global pool)

Memory and Performance Considerations

Reduction Performance:
┌─────────────────┬──────────────┬─────────────────┬─────────────────┐
│ Operation       │ Time Complex │ Memory Access   │ Cache Behavior  │
├─────────────────┼──────────────┼─────────────────┼─────────────────┤
│ .sum()          │ O(N)         │ Sequential read │ Excellent       │
│ .sum(axis=0)    │ O(N)         │ Column access   │ Poor (strided)  │
│ .sum(axis=1)    │ O(N)         │ Row access      │ Excellent       │
│ .mean()         │ O(N)         │ Sequential read │ Excellent       │
│ .max()          │ O(N)         │ Sequential read │ Excellent       │
└─────────────────┴──────────────┴─────────────────┴─────────────────┘

Why axis=0 is slower:
- Accesses elements with large strides
- Poor cache locality (jumping rows)
- Less vectorization-friendly

Optimization strategies:
- Prefer axis=-1 operations when possible
- Use keepdims=True to maintain shape for broadcasting
- Consider reshaping before reduction for better cache behavior

🧪 Unit Test: Reduction Operations

This test validates reduction operations work correctly with axis control and maintain proper shapes.

What we're testing: Sum, mean, max operations with axis parameter and keepdims Why it matters: Essential for loss computation, batch processing, and pooling operations Expected: Correct reduction along specified axes with proper shape handling

In [ ]:

def test_unit_reduction_operations():
    """🧪 Test reduction operations."""
    print("🧪 Unit Test: Reduction Operations...")

    matrix = Tensor([[1, 2, 3], [4, 5, 6]])  # Shape: (2, 3)

    # Test sum all elements (common for loss computation)
    total = matrix.sum()
    assert total.data == 21.0  # 1+2+3+4+5+6
    assert total.shape == ()   # Scalar result

    # Test sum along axis 0 (columns) - batch dimension reduction
    col_sum = matrix.sum(axis=0)
    expected_col = np.array([5, 7, 9], dtype=np.float32)  # [1+4, 2+5, 3+6]
    assert np.array_equal(col_sum.data, expected_col)
    assert col_sum.shape == (3,)

    # Test sum along axis 1 (rows) - feature dimension reduction
    row_sum = matrix.sum(axis=1)
    expected_row = np.array([6, 15], dtype=np.float32)  # [1+2+3, 4+5+6]
    assert np.array_equal(row_sum.data, expected_row)
    assert row_sum.shape == (2,)

    # Test mean (average loss computation)
    avg = matrix.mean()
    assert np.isclose(avg.data, 3.5)  # 21/6
    assert avg.shape == ()

    # Test mean along axis (batch normalization pattern)
    col_mean = matrix.mean(axis=0)
    expected_mean = np.array([2.5, 3.5, 4.5], dtype=np.float32)  # [5/2, 7/2, 9/2]
    assert np.allclose(col_mean.data, expected_mean)

    # Test max (finding best predictions)
    maximum = matrix.max()
    assert maximum.data == 6.0
    assert maximum.shape == ()

    # Test max along axis (argmax-like operation)
    row_max = matrix.max(axis=1)
    expected_max = np.array([3, 6], dtype=np.float32)  # [max(1,2,3), max(4,5,6)]
    assert np.array_equal(row_max.data, expected_max)

    # Test keepdims (important for broadcasting)
    sum_keepdims = matrix.sum(axis=1, keepdims=True)
    assert sum_keepdims.shape == (2, 1)  # Maintains 2D shape
    expected_keepdims = np.array([[6], [15]], dtype=np.float32)
    assert np.array_equal(sum_keepdims.data, expected_keepdims)

    # Test 3D reduction (simulating global average pooling)
    tensor_3d = Tensor([[[1, 2], [3, 4]], [[5, 6], [7, 8]]])  # (2, 2, 2)
    spatial_mean = tensor_3d.mean(axis=(1, 2))  # Average across spatial dimensions
    assert spatial_mean.shape == (2,)  # One value per batch item

    print("✅ Reduction operations work correctly!")

if __name__ == "__main__":
    test_unit_reduction_operations()

Gradient Features: Preparing for Module 05

Our Tensor includes dormant gradient features that will spring to life in Module 05. For now, they exist but do nothing - this design choice ensures a consistent interface throughout the course.

Why Include Gradient Features Now?

Gradient System Evolution:
Module 01: Tensor with dormant gradients
  ┌─────────────────────────────────┐
  │ Tensor                          │
  │ • data: actual values          │
  │ • requires_grad: False         │ ← Present but unused
  │ • grad: None                   │ ← Present but stays None
  │ • backward(): pass             │ ← Present but does nothing
  └─────────────────────────────────┘
         ↓ Module 05 activates these
Module 05: Tensor with active gradients
  ┌─────────────────────────────────┐
  │ Tensor                          │
  │ • data: actual values          │
  │ • requires_grad: True          │ ← Now controls gradient tracking
  │ • grad: computed gradients     │ ← Now accumulates gradients
  │ • backward(): computes grads   │ ← Now implements chain rule
  └─────────────────────────────────┘

Design Benefits

Consistency: Same Tensor class interface throughout all modules

• No confusing Variable vs. Tensor distinction (unlike early PyTorch)
• Students never need to learn a "new" Tensor class
• IDE autocomplete works from day one

Gradual Complexity: Features activate when students are ready

• Module 01-04: Ignore gradient features, focus on operations
• Module 05: Gradient features "turn on" magically
• No cognitive overload in early modules

Future-Proof: Easy to extend without breaking changes

• Additional features can be added as dormant initially
• No monkey-patching or dynamic class modification
• Clean evolution path

Current State (Module 01)

Gradient Features - Current Behavior:
┌─────────────────────────────────────────────────────────┐
│ Feature           │ Current State  │ Module 05 State    │
├─────────────────────────────────────────────────────────┤
│ requires_grad     │ False          │ True (when needed) │
│ grad              │ None           │ np.array(...)      │
│ backward()        │ pass (no-op)   │ Chain rule impl    │
│ Operation chaining│ Not tracked    │ Computation graph  │
└─────────────────────────────────────────────────────────┘

Student Experience:
• Can call .backward() without errors (just does nothing)
• Can set requires_grad=True (just gets stored)
• Focus on understanding tensor operations first
• Gradients remain "mysterious" until Module 05 reveals them

This approach matches the pedagogical principle of "progressive disclosure" - reveal complexity only when students are ready to handle it.

4. Integration: Bringing It Together

Let's test how our Tensor operations work together in realistic scenarios that mirror neural network computations. This integration demonstrates that our individual operations combine correctly for complex ML workflows.

Neural Network Layer Simulation

The fundamental building block of neural networks is the linear transformation: y = xW + b

Linear Layer Forward Pass: y = xW + b

Input Features → Weight Matrix → Matrix Multiply → Add Bias → Output Features
  (batch, in)   (in, out)        (batch, out)     (batch, out)   (batch, out)

Step-by-Step Breakdown:
1. Input:   X shape (batch_size, input_features)
2. Weight:  W shape (input_features, output_features)
3. Matmul:  XW shape (batch_size, output_features)
4. Bias:    b shape (output_features,)
5. Result:  XW + b shape (batch_size, output_features)

Example Flow:
Input: [[1, 2, 3],    Weight: [[0.1, 0.2],    Bias: [0.1, 0.2]
        [4, 5, 6]]            [0.3, 0.4],
       (2, 3)                 [0.5, 0.6]]
                             (3, 2)

Step 1: Matrix Multiply
[[1, 2, 3]] @ [[0.1, 0.2]] = [[1×0.1+2×0.3+3×0.5, 1×0.2+2×0.4+3×0.6]]
[[4, 5, 6]]   [[0.3, 0.4]]   [[4×0.1+5×0.3+6×0.5, 4×0.2+5×0.4+6×0.6]]
              [[0.5, 0.6]]
                           = [[1.6, 2.6],
                              [4.9, 6.8]]

Step 2: Add Bias (Broadcasting)
[[1.6, 2.6]] + [0.1, 0.2] = [[1.7, 2.8],
 [4.9, 6.8]]                 [5.0, 7.0]]

This is the foundation of every neural network layer!

Why This Integration Matters

This simulation shows how our basic operations combine to create the computational building blocks of neural networks:

• Matrix Multiplication: Transforms input features into new feature space
• Broadcasting Addition: Applies learned biases efficiently across batches
• Shape Handling: Ensures data flows correctly through layers
• Memory Management: Creates new tensors without corrupting inputs

Every layer in a neural network - from simple MLPs to complex transformers - uses this same pattern.

"""

🧪 Module Integration Test

Final validation that everything works together correctly before module completion. """

def import_previous_module(module_name: str, component_name: str): import sys import os sys.path.append(os.path.join(os.path.dirname(file), '..', module_name)) module = import(f"{module_name.split('_')[1]}_dev") return getattr(module, component_name)

In [ ]:

def test_module():
    """
    Comprehensive test of entire module functionality.

    This final test runs before module summary to ensure:
    - All unit tests pass
    - Functions work together correctly
    - Module is ready for integration with TinyTorch
    """
    print("🧪 RUNNING MODULE INTEGRATION TEST")
    print("=" * 50)

    # Run all unit tests
    print("Running unit tests...")
    test_unit_tensor_creation()
    test_unit_arithmetic_operations()
    test_unit_matrix_multiplication()
    test_unit_shape_manipulation()
    test_unit_reduction_operations()

    print("\nRunning integration scenarios...")

    # Test realistic neural network computation
    print("🧪 Integration Test: Two-Layer Neural Network...")

    # Create input data (2 samples, 3 features)
    x = Tensor([[1, 2, 3], [4, 5, 6]])

    # First layer: 3 inputs → 4 hidden units
    W1 = Tensor([[0.1, 0.2, 0.3, 0.4],
                 [0.5, 0.6, 0.7, 0.8],
                 [0.9, 1.0, 1.1, 1.2]])
    b1 = Tensor([0.1, 0.2, 0.3, 0.4])

    # Forward pass: hidden = xW1 + b1
    hidden = x.matmul(W1) + b1
    assert hidden.shape == (2, 4), f"Expected (2, 4), got {hidden.shape}"

    # Second layer: 4 hidden → 2 outputs
    W2 = Tensor([[0.1, 0.2], [0.3, 0.4], [0.5, 0.6], [0.7, 0.8]])
    b2 = Tensor([0.1, 0.2])

    # Output layer: output = hiddenW2 + b2
    output = hidden.matmul(W2) + b2
    assert output.shape == (2, 2), f"Expected (2, 2), got {output.shape}"

    # Verify data flows correctly (no NaN, reasonable values)
    assert not np.isnan(output.data).any(), "Output contains NaN values"
    assert np.isfinite(output.data).all(), "Output contains infinite values"

    print("✅ Two-layer neural network computation works!")

    # Test gradient attributes are preserved and functional
    print("🧪 Integration Test: Gradient System Readiness...")
    grad_tensor = Tensor([1, 2, 3], requires_grad=True)
    result = grad_tensor + 5
    assert grad_tensor.requires_grad == True, "requires_grad not preserved"
    assert grad_tensor.grad is None, "grad should still be None"

    # Test backward() doesn't crash (even though it does nothing)
    grad_tensor.backward()  # Should not raise any exception

    print("✅ Gradient system ready for Module 05!")

    # Test complex shape manipulations
    print("🧪 Integration Test: Complex Shape Operations...")
    data = Tensor([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12])

    # Reshape to 3D tensor (simulating batch processing)
    tensor_3d = data.reshape(2, 2, 3)  # (batch=2, height=2, width=3)
    assert tensor_3d.shape == (2, 2, 3)

    # Global average pooling simulation
    pooled = tensor_3d.mean(axis=(1, 2))  # Average across spatial dimensions
    assert pooled.shape == (2,), f"Expected (2,), got {pooled.shape}"

    # Flatten for MLP
    flattened = tensor_3d.reshape(2, -1)  # (batch, features)
    assert flattened.shape == (2, 6)

    # Transpose for different operations
    transposed = tensor_3d.transpose()  # Should transpose last two dims
    assert transposed.shape == (2, 3, 2)

    print("✅ Complex shape operations work!")

    # Test broadcasting edge cases
    print("🧪 Integration Test: Broadcasting Edge Cases...")

    # Scalar broadcasting
    scalar = Tensor(5.0)
    vector = Tensor([1, 2, 3])
    result = scalar + vector  # Should broadcast scalar to vector shape
    expected = np.array([6, 7, 8], dtype=np.float32)
    assert np.array_equal(result.data, expected)

    # Matrix + vector broadcasting
    matrix = Tensor([[1, 2], [3, 4]])
    vec = Tensor([10, 20])
    result = matrix + vec
    expected = np.array([[11, 22], [13, 24]], dtype=np.float32)
    assert np.array_equal(result.data, expected)

    print("✅ Broadcasting edge cases work!")

    print("\n" + "=" * 50)
    print("🎉 ALL TESTS PASSED! Module ready for export.")
    print("Run: tito module complete 01_tensor")

# Run comprehensive module test
if __name__ == "__main__":
    test_module()

🤔 ML Systems Assessment Questions

Before completing this module, test your understanding with these quantitative problems. These questions help consolidate your knowledge and prepare you for production ML engineering.

Question 1: Memory Requirements (3 points)

Calculate the memory required for these tensors in float32:

• Tensor A: (1000, 1000)
• Tensor B: (500, 2000)

TODO: Fill in your calculations below with units (MB or GB)

APPROACH:

1. Calculate total elements: rows × columns
2. Multiply by bytes per element (float32 = 4 bytes)
3. Convert to MB (divide by 1024²)
4. Compare memory usage

In [ ]:

# YOUR ANSWER:
#
# Tensor A (1000, 1000) in float32:
# - Elements: ___________
# - Memory: ___________ MB
#
# Tensor B (500, 2000) in float32:
# - Elements: ___________
# - Memory: ___________ MB
#
# Which uses more memory? ___________
# How much more? ___________ MB

### BEGIN SOLUTION
# Tensor A: 1000 × 1000 = 1,000,000 elements
# Memory: 1,000,000 × 4 bytes = 4,000,000 bytes = 3.81 MB

# Tensor B: 500 × 2000 = 1,000,000 elements
# Memory: 1,000,000 × 4 bytes = 4,000,000 bytes = 3.81 MB

# Answer: Same memory usage (both have 1M elements)
# Difference: 0 MB - shape doesn't matter, only total elements
### END SOLUTION

Question 2: Computational Complexity (3 points)

Calculate FLOPs for a 3-layer neural network:

• Layer 1: Input (batch=64, features=784) → Hidden (batch=64, features=256)
• Layer 2: Hidden (batch=64, features=256) → Hidden (batch=64, features=128)
• Layer 3: Hidden (batch=64, features=128) → Output (batch=64, features=10)

TODO: Calculate total FLOPs for one forward pass

HINT: For matrix multiplication (M,K) @ (K,N), FLOPs = 2 × M × K × N

In [ ]:

# YOUR ANSWER:
#
# Layer 1 FLOPs: ___________
# Layer 2 FLOPs: ___________
# Layer 3 FLOPs: ___________
# Total FLOPs: ___________ (in millions)

### BEGIN SOLUTION
# Layer 1: (64, 784) @ (784, 256)
# FLOPs = 2 × 64 × 784 × 256 = 25,690,112

# Layer 2: (64, 256) @ (256, 128)
# FLOPs = 2 × 64 × 256 × 128 = 4,194,304

# Layer 3: (64, 128) @ (128, 10)
# FLOPs = 2 × 64 × 128 × 10 = 163,840

# Total: 25,690,112 + 4,194,304 + 163,840 = 30,048,256 FLOPs
# ≈ 30 million FLOPs per forward pass
### END SOLUTION

Question 3: Broadcasting Behavior (2 points)

Predict the output shape for these operations:

A = Tensor with shape (32, 64)    # Matrix
B = Tensor with shape (64,)       # Vector
C = Tensor with shape (32, 1)     # Column vector
D = Tensor with shape (1, 64)     # Row vector

TODO: Fill in the resulting shapes

HINT: Broadcasting aligns from the right, dimensions must match or be 1

In [ ]:

# YOUR ANSWER:
#
# A + B → Shape: ___________
# A + C → Shape: ___________
# A + D → Shape: ___________
# B + C → Shape: ___________
# C + D → Shape: ___________

### BEGIN SOLUTION
# A + B: (32, 64) + (64,) → (32, 64)  [broadcast B to each row]
# A + C: (32, 64) + (32, 1) → (32, 64)  [broadcast C to each column]
# A + D: (32, 64) + (1, 64) → (32, 64)  [broadcast D to each row]
# B + C: (64,) + (32, 1) → (32, 64)  [both broadcast to 2D]
# C + D: (32, 1) + (1, 64) → (32, 64)  [outer product-like broadcast]
### END SOLUTION

Question 4: Production Scaling (2 points)

A neural network layer has shape (batch, 512) @ (512, 1024).

TODO: Answer these scaling questions

1. If batch size doubles from 32 to 64, how do FLOPs scale?
2. If we use float16 instead of float32, how does memory scale?
3. What's the performance bottleneck: computation or memory bandwidth?

In [ ]:

# YOUR ANSWER:
#
# 1. FLOPs scaling when batch doubles: ___________
#    (same / 2× / 4× / 8×?)
#
# 2. Memory scaling with float16 vs float32: ___________
#    (same / 0.5× / 0.25× / 2×?)
#
# 3. Performance bottleneck: ___________
#    (computation / memory bandwidth / both?)
#
# Reasoning: ___________

### BEGIN SOLUTION
# 1. FLOPs scale linearly with batch size: 2× FLOPs
#    Original: 2 × 32 × 512 × 1024 = 33,554,432 FLOPs
#    Doubled: 2 × 64 × 512 × 1024 = 67,108,864 FLOPs (2× increase)

# 2. Memory scales with precision: 0.5× memory (half the bytes per element)
#    float32: 4 bytes/element
#    float16: 2 bytes/element (50% reduction)

# 3. Bottleneck: Memory bandwidth for large batch sizes
#    - Modern GPUs have high FLOP/s (teraFLOPs)
#    - Memory bandwidth is limited (100s of GB/s)
#    - Large matrices → more data movement than computation
#    - For small batches: computation bound
#    - For large batches: memory bandwidth bound
### END SOLUTION

🎯 MODULE SUMMARY: Tensor Foundation

Congratulations! You've built the foundational Tensor class that powers all machine learning operations!

Key Accomplishments

• Built a complete Tensor class with arithmetic operations, matrix multiplication, and shape manipulation
• Implemented broadcasting semantics that match NumPy for automatic shape alignment
• Created dormant gradient features that will activate in Module 05 (autograd)
• Added comprehensive ASCII diagrams showing tensor operations visually
• All methods defined INSIDE the class (no monkey-patching) for clean, maintainable code
• All tests pass ✅ (validated by test_module())

Systems Insights Discovered

• Memory scaling: Matrix operations create new tensors (3× memory during computation)
• Broadcasting efficiency: NumPy's automatic shape alignment vs. explicit operations
• Shape validation trade-offs: Clear errors vs. performance in tight loops
• Architecture decisions: Dormant features vs. inheritance for clean evolution

Ready for Next Steps

Your Tensor implementation enables all future modules! The dormant gradient features will spring to life in Module 05, and every neural network component will build on this foundation.

Export with: tito module complete 01_tensor

Next: Module 02 will add activation functions (ReLU, Sigmoid, GELU) that bring intelligence to neural networks by introducing nonlinearity!