Offering insights across various domains such as computer vision and natural language processing, Vectorization covers the fundamental topics of vectorization including array and tensor operations, data wrangling, and batch processing. This book illustrates how the principles discussed lead to successful outcomes in machine learning projects, serving as concrete examples for the theories explained, with each chapter including practical case studies and code implementations using NumPy, TensorFlow, and PyTorch.
Each chapter has one or two types of contents: either an introduction/comparison of the specific operations in the numerical libraries (illustrated as tables) and/or case study examples that apply the concepts introduced to solve a practical problem (as code blocks and figures). Readers can approach the knowledge presented by reading the text description, running the code blocks, or examining the figures.
Written by the developer of the first recommendation system on the Peacock streaming platform, Vectorization explores sample topics including: - Basic tensor operations and the art of tensor indexing, elucidating how to access individual or subsets of tensor elements- Vectorization in tensor multiplications and common linear algebraic routines, which form the backbone of many machine learning algorithms- Masking and padding, concepts which come into play when handling data of non-uniform sizes, and string processing techniques for natural language processing (NLP)- Sparse matrices and their data structures and integral operations, and ragged or jagged tensors and the nuances of processing them
From the essentials of vectorization to the subtleties of advanced data structures, Vectorization is an ideal one-stop resource for both beginners and experienced practitioners, including researchers, data scientists, statisticians, and other professionals in industry, who seek academic success and career advancement.
Table of Contents
About the Author xiii
Preface xv
Acknowledgment xix
1 Introduction to Vectorization 1
1.1 What Is Vectorization 1
1.1.1 A Simple Example of Vectorization in Action 2
1.1.2 Python Can Still Be Faster! 3
1.1.3 Memory Allocation of Vectorized Operations 4
1.2 Case Study: Dense Layer of a Neural Network 6
1.3 Vectorization vs. Other Parallel Computing Paradigms 9
1.3.1 Multithreading 9
1.3.2 Multiprocessing 9
1.3.3 Multiworker Distributed Computing 13
Bibliography 16
2 Basic Tensor Operations 19
2.1 Tensor Initializers 19
2.2 Data Type and Casting 24
2.2.1 Tips on Specifying the dtypes During Tensor Initialization 27
2.2.2 Tips on Casting 27
2.3 Mathematical Operations 27
2.4 Reduction Operations 31
2.5 Value Comparison Operations 31
2.6 Logical Operations 32
2.7 Ordered Array-Adjacent Element Operations 33
2.8 Array Reversing 33
2.9 Concatenation, Stacking, and Splitting 35
2.10 Reshaping 35
2.11 Broadcasting 38
2.12 Case Studies 44
2.12.1 Image Normalization 45
2.12.2 Pearson’s Correlation 46
2.12.3 Pair-wise Difference 47
2.12.4 Construction of Magic Squares 48
Bibliography 57
3 Tensor Indexing 61
3.1 Get Values at Index 61
3.1.1 Integer Indexing 61
3.1.2 Flat Index vs. Multi-index 63
3.1.3 Boolean Indexing 69
3.2 Slicing 70
3.2.1 Reusing Slice Configuration 75
3.3 Case Study: Get Consecutive Index 78
3.4 Take and Gather 80
3.4.1 Take 80
3.4.2 Take Along Axis 83
3.4.3 Gather 87
3.4.4 N-Dimensional Gather 91
3.5 Assign Values at Index 95
3.6 Put and Scatter 98
3.6.1 Put 98
3.6.2 Put Along Axis 100
3.6.3 Multi-index Scatter Replacement 101
3.6.4 Additional Scatter Operations from PyTorch 108
3.7 Case Study: Batchwise Scatter Values 113
Bibliography 115
4 Linear Algebra 119
4.1 Tensor Multiplications 119
4.2 The matmul Operation 119
4.2.1 The @ Operator 122
4.3 The tensordot Operation 123
4.3.1 Heuristics of tensordot Operations 125
4.4 Einsum 129
4.5 Case Study: Pair-wise Pearson’s Cross-Correlation 134
4.6 Case Study: Hausdorff Distance 135
4.7 Common Linear Algebraic Routines 139
4.8 Case Study: Fitting Single Exponential Curves 139
Bibliography 144
5 Masking and Padding 145
5.1 Masking 145
5.1.1 Triangular and Diagonal Masks 146
5.1.2 Changing Elements Using the where Operation 146
5.1.3 Use Multiplication to Apply Masks 146
5.1.4 Use Arithmetic Operations as Boolean Operations to Apply and Combine Masks 151
5.1.5 Select Elements Based on Masking 152
5.1.6 Case Study: Top-k Masking 153
5.2 Padding 155
5.2.1 Case Study: Padding in Convolutional Neural Networks 161
5.2.2 Case Study: Truncate or Pad Sequence to Desired Length 163
5.3 Advanced Case Studies 164
5.3.1 Scaled-Dot Product Attention 164
5.3.2 Variable-Length Range via Masking 168
5.3.3 Length Regulator Module of FastSpeech 2 171
Bibliography 181
6 String Processing 183
6.1 String Data Types 183
6.1.1 NumPy String, Bytes, and Object 183
6.1.2 Pandas String 184
6.1.3 Tensorflow Bytes 186
6.1.4 PyTorch 187
6.2 String Operations 187
6.3 Case Study: Parsing DateTime from String Representations 189
6.4 Mapping Strings to Indices 194
6.4.1 NumPy np.unique 194
6.4.2 Pandas pd.Categorical 195
6.4.3 Scikit-learn sklearn.preprocessing.LabelEncoder 198
6.4.4 Tensorflow tf.lookup 198
6.4.5 TorchText torchtext.vocab 200
6.5 Case Study: Factorization Machine 201
6.5.1 Factorization Machine Model 202
6.5.2 More Efficient Optimization Criterion 202
6.5.3 Implementation of Deep Factorization Machine in Tensorflow 203
6.5.4 Training DeepFM on MovieLens 1M Dataset 209
6.6 Regular Expressions (Regex) 215
6.7 Data Serialization and Deserialization 217
Bibliography 221
7 Sparse Matrix 223
7.1 Scipy’s Sparse Matrix Classes 224
7.1.1 Coordinate Sparse Matrix (coo_matrix) 224
7.1.2 Compressed Sparse Column Matrix (csc_matrix) 225
7.1.3 Compressed Sparse Row Matrix (csr_matrix) 227
7.1.4 Block Sparse Row Matrix (bsr_matrix) 228
7.1.5 Dictionary of Keys Sparse Matrix (dok_matrix) 229
7.1.6 Row-Based List of List Sparse Matrix (lil_matrix) 230
7.1.7 Diagonal Storage Sparse Matrix (dia_matrix) 232
7.1.8 Comparisons Between Different Sparse Matrix Formats 233
7.2 Sparse Matrix Broadcasting 235
7.2.1 Scalar Broadcasting 235
7.2.2 Row-wise Broadcasting 236
7.2.3 Column-wise Broadcasting 237
7.2.4 Multiplication on Sparse Indices 237
7.3 Tensorflow’s Sparse Tensors 238
7.3.1 SparseTensor Class 239
7.3.2 Sparse CSR Matrix 240
7.4 PyTorch’s Sparse Matrix 242
7.5 Sparse Matrix in Other Python Libraries 245
7.6 When (Not) to Use Sparse Matrix 245
7.7 Case Study: Sparse Matrix Factorization with ALS 245
7.7.1 Matrix Factorization 246
7.7.2 Parameter Updates with ALS 246
7.7.3 Adding Bias Terms to Matrix Factorization 247
7.7.4 Adding Regularization Term 249
7.7.5 Implementing ALS 250
7.7.6 Training a Model with MovieLens-100k 255
Bibliography 257
8 Jagged Tensors 261
8.1 Left Align a Sparse Tensor to Represent Ragged Tensor 263
8.2 Index to Binary Indicator 269
8.3 Case Study: Jaccard Similarities Using Sparse Matrix 271
8.4 Case Study: Batchwise Set Operations 275
8.5 Case Study: Autoencoders with Sparse Inputs 283
8.5.1 Embedding Lookup on Sparse Inputs 284
8.5.2 Inputs with Weights 287
Bibliography 293
9 Groupby, Apply, and Aggregate 295
9.1 Pandas Groupwise Operations 296
9.2 Reshaping and Windowing of Dense Tensors 298
9.3 Case Study: Vision Transformer (ViT) 305
9.4 Bucketizing Values 315
9.5 Segment-wise Aggregation 319
9.6 Case Study: EmbeddingBag 325
9.7 Case Study: Vocal Duration Constrained by Note Duration 330
9.8 Case Study: Filling of Missing Values in a Sequence 336
Bibliography 341
10 Sorting and Reordering 343
10.1 Sorting Operations 343
10.2 Case Study: Top-k Using argsort and argpartition 346
10.3 Case Study: Sort the Rows of a Matrix 349
10.4 Case Study: Reverse Padded Sequence 353
10.5 Case Study: Gumbel-Max Sampling with Weights 358
10.6 Case Study: Sorting Articles Around Anchored Advertisements 367
Bibliography 373
11 Building a Language Model from Scratch 375
11.1 Language Modeling with Transformer 375
11.1.1 Encoder and Decoder of the Transformer Architecture 376
11.1.2 Training of Transformer Models 377
11.2 Pre-LN vs. Post-LN Transformer 378
11.3 Layer Normalization 383
11.4 Positional Encoding and Embedding 385
11.4.1 Sinusoidal Positional Encoding 385
11.4.2 Position as Categorical Embeddings 387
11.4.3 Relative Positional Encoding (RPE) 388
11.4.4 Rotary Positional Encoding (RoPE) 389
11.5 Activation Functions in Feedforward Layer 395
11.6 Case Study: Training a Tiny LLaMA Model for Next Token Prediction 398
11.7 A Word on AI Safety and Alignment 410
11.8 Concluding Remarks 412
Bibliography 412
Index 419
