Dyads

Definition of dyads

• Dyad: Matrix $A\in {\mathbb{R}}^{m,n}$ that can be written in the form $A=p{q}^T$
  • $p\in {\mathbb{R}}^m$, $q\in {\mathbb{R}}^n$
  • interpretation: here
• Individual element: $A_{ij}=p_i.q_j$
  • $A_{ij}$ is the element in the (i)th row and (j)th column of the matrix A
  • $p_i$: i-th element of p, $(1\le i\le m)$
  • $q_j$: j-th element of q, $(1\le j\le n)$
• Interpretation:
  • the columns of p are scaled, by a scaling factor given in vector q, and result in the columns in A
  • the rows of $q^T$ are scaled, by a scaling factor given in vector p, and result in the rows in A
• Examples of video frames
  • a set of image frames representing a video, each image represented by a row vector of pixels → can represent the whole video as a matrix A
  • if the video shows a no-movement scene → each row in matrix A is identical → all rows are scaled copies of the same row vector + all cols are scaled copies of the same col vector → matrix A is dyad

Sum of dyads

• SVD theorem: any matrix can be written as a sum of dyads: $A={\sum }_{i=1}^r{p}_i{q}_i^T$ ($p_i,q_i$ are mutually orthogonal)
• can interpret data as sum of simpler matrix (dyads)
• mutual orthogonality of each dyad ensures each encodes independent information