# Low-rank approximation 2/4

Tech

Overview of Low-rank approximation – Part2

### 1. **Matrix Decomposition Techniques**

**Singular Value Decomposition (SVD)**: SVD is a linear algebra technique that decomposes a matrix into three other matrices. For a convolutional filter represented as a matrix \(W\), SVD finds matrices \(U\), \(\Sigma\), and \(V\) such that \(W = U \Sigma V^T\).**Application**: By keeping only the top \(k\) singular values from \(\Sigma\), and the corresponding columns in \(U\) and \(V\), we can create a low-rank approximation of the original filter.**Benefits**: This significantly reduces the number of parameters and computations involved.

**Tucker Decomposition**: Tucker decomposition generalizes SVD for tensors (multi-dimensional arrays). It decomposes a tensor into a core tensor and a matrix factorization of each mode (dimension).**Application**: In CNNs, filters can be represented as 3D tensors (width, height, depth). Tucker decomposition approximates the filter as a smaller core tensor with smaller matrices for each dimension.**Benefits**: Allows a more flexible approximation with potentially better representation capabilities compared to SVD.

**CP (CANDECOMP/PARAFAC) Decomposition**: CP decomposition also approximates a tensor by expressing it as the sum of a finite number of rank-one tensors.**Application**: In CNNs, it can help decompose filters into simpler components that capture the same information with fewer parameters.**Benefits**: It can be simpler to implement in certain situations than Tucker decomposition.

### 2. **Filter Factorization Techniques**

**Two-Dimensional Factorization**: Instead of using one large convolutional kernel, it decomposes it into two smaller kernels, typically one for spatial dimensions (width and height) and one for the depth dimension.**Application**: A \(k \times k \times d\) filter can be approximated by multiplying a \(k \times k\) filter with a \(k \times 1\) filter for each depth slice.**Benefits**: This reduces the computational complexity from \(O(k^2 \cdot d)\) to \(O(k^2 + k \cdot d)\).

**Depthwise Separable Convolutions**: This technique separates the convolution process into two layers: a depthwise convolution (applies a single filter per input channel) followed by a pointwise convolution (1x1 convolution to combine the outputs).**Application**: MobileNets, for instance, use this technique extensively to reduce the number of parameters while maintaining performance.**Benefits**: It reduces both computational and memory requirements significantly compared to standard convolutions.

### 3. **Quantization Techniques**

**Weight Quantization**: Involves reducing the precision of the weights (e.g., using 8-bit integers instead of 32-bit floating-point numbers).**Application**: After performing low-rank approximation, quantization can further reduce the model size and speed up inference on hardware with limited precision.**Benefits**: Lower memory usage and faster computation, especially on specialized hardware like GPUs or TPUs.

### 4. **Structured Low-Rank Approximations**

**Low-Rank Convolutional Filters**: Instead of fully dense filters, structured filters with fewer connections can be designed to maintain performance while reducing rank.**Application**: This may involve using predefined patterns or structures for the filters based on the spatial hierarchies present in the data.**Benefits**: More efficient use of parameters while exploiting spatial correlation in images.

### 5. **Regularization Techniques**

**Low-Rank Regularization**: Adding a low-rank regularization term to the loss function during training encourages the network to learn low-rank filters.**Application**: This could be achieved by penalizing the nuclear norm (sum of singular values) of the filter matrices in the loss function.**Benefits**: Helps in obtaining low-rank approximations directly from the training process, maintaining a good trade-off between accuracy and model size.