10.2 Tensor operations and decompositions

Tensors take data analysis to new dimensions, literally. They're like super-powered matrices, handling complex relationships in high-dimensional data. This section dives into the nuts and bolts of tensor operations, from basic addition to advanced decomposition techniques.

We'll explore how to manipulate tensors, extract patterns, and reduce dimensionality. These tools are crucial for tackling real-world problems in areas like image processing, recommender systems, and brain imaging. Get ready to level up your data science toolkit!

Tensor operations

Addition and multiplication

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• Tensors generalize vectors and matrices to higher dimensions representing complex data relationships
• Tensor addition involves element-wise addition of tensors with the same shape
• Tensor multiplication types
  • Element-wise multiplication
  • Tensor-vector multiplication
  • Tensor-matrix multiplication
  • Tensor-tensor multiplication
• Broadcasting expands smaller tensors to match larger ones for operations on tensors with different shapes
• Examples:
  • Element-wise addition: $A_{ijk} + B_{ijk} = C_{ijk}$
  • Broadcasting: Adding a vector to each slice of a 3D tensor

Contraction and notation

• Tensor contraction sums over one or more indices reducing dimensionality
• Einstein summation convention concisely expresses tensor operations particularly contractions
• Contraction produces a lower-order tensor or scalar
• Examples:
  • Matrix-vector multiplication as tensor contraction: $y_i = A_{ij}x_j$
  • Trace of a matrix: $A_{ii}$ (sum over repeated index)

Tensor rank and unfolding

Rank concepts

• Tensor rank generalizes matrix rank to higher dimensions
• CP rank (CANDECOMP/PARAFAC rank) represents minimum rank-one components for exact tensor representation
• Tucker rank provides a multidimensional perspective on tensor complexity
• Tucker rank expressed as a tuple of ranks of different matricizations
• Examples:
  • CP rank of a rank-one tensor: 1
  • Tucker rank of a 3D tensor: (r1, r2, r3) where ri is the rank of mode-i unfolding

Unfolding techniques

• Tensor unfolding (matricization or flattening) reshapes a tensor into a matrix preserving elements
• Mode-n unfolding arranges n-th mode fibers as matrix columns
• Multiple unfolding methods exist for a given tensor
• Unfolding crucial for analyzing tensor structure and applying decomposition techniques
• Examples:
  • Mode-1 unfolding of a 3D tensor: Arranging each horizontal slice as a column
  • Mode-2 unfolding of a 3D tensor: Arranging each vertical slice as a column

Tensor decomposition for pattern extraction

Dimensionality reduction methods

• Tensor decomposition generalizes matrix factorization to higher-order tensors
• CANDECOMP/PARAFAC (CP) decomposition approximates tensor as sum of rank-one tensors
• Tucker decomposition (higher-order SVD) factorizes tensor into core tensor multiplied by factor matrices
• Tensor Train (TT) decomposition represents high-order tensor as chain of lower-order tensors
• Examples:
  • CP decomposition: $X \approx \sum_{r=1}^R a_r \circ b_r \circ c_r$ for a 3D tensor
  • Tucker decomposition: $X \approx G \times_1 A \times_2 B \times_3 C$ where G is the core tensor

Advanced techniques

• Non-negative tensor factorization (NTF) imposes non-negativity constraints on decomposition factors
• NTF useful for applications where negative values lack meaning (image processing)
• Tensor decomposition reveals latent structures identifies underlying patterns and compresses high-dimensional data
• Examples:
  • NTF in spectral data analysis: Decomposing chemical spectra into non-negative components
  • Tensor decomposition in EEG analysis: Extracting spatial temporal and spectral features

Tensor decomposition methods vs applications

Method characteristics

• CP decomposition extracts interpretable components and handles sparse data
• CP decomposition may suffer from degeneracy issues in some cases
• Tucker decomposition offers flexibility in modeling interactions between different modes
• Tucker decomposition may require more storage for the core tensor
• Tensor Train decomposition efficiently handles very high-dimensional data
• HOSVD extends matrix SVD to tensors but may not yield optimal low-rank approximation
• Examples:
  • CP decomposition in chemometrics: Analyzing chemical mixtures
  • Tucker decomposition in computer vision: Analyzing facial expressions across individuals poses and lighting conditions

Application considerations

• Decomposition method choice depends on application data characteristics and desired trade-offs
• Trade-offs include interpretability computational efficiency and approximation accuracy
• Applications span signal processing computer vision recommender systems and spatiotemporal data analysis
• Evaluation metrics include reconstruction error computational complexity and component interpretability
• Examples:
  • Tensor decomposition in recommender systems: Modeling user-item-context interactions
  • Tensor-based analysis of fMRI data: Extracting spatial temporal and subject-specific patterns in brain activity