# Latent Semantic Analysis Uses [[Singular Value Decomposition (SVD)]] to create representations of documents. $ \begin{array}{c} \left(\mathbf{t}_{i}^{T}\right) \rightarrow\left[\begin{array}{ccc} x_{1,1} & \cdots & x_{1, n} \\ \vdots & \ddots & \vdots \\ x_{m, 1} & \ldots & x_{m, n} \end{array}\right]=\left(\hat{\mathbf{t}}_{i}^{T}\right) \rightarrow\left[\left[\mathbf{u}_{1}\right] \ldots\left[\mathbf{u}_{k}\right]\right] \cdot\left[\begin{array}{ccc} \sigma_{1} & \ldots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \ldots & \sigma_{k} \end{array}\right] \cdot\left[\begin{array}{c} {\left[\begin{array}{c} \mathbf{v}_{1} \end{array}\right]} \\ \vdots \\ {\left[\begin{array}{c} \mathbf{v}_{k} \end{array}\right]} \end{array}\right] \\ d_{j}=U_{k} \Sigma_{k} \hat{d}_{j} \Longrightarrow \hat{d}_{j}=\Sigma_{k}^{-1} U_{k}^{T} d_{j} \end{array} $ Given a collection of documents, perform SVD and low-rank approximation to obtain $\Sigma_{k}$ and $U_{k}$ Given a document and a query, represent them as a vectors in the obtained "semantic" vector space $ \begin{array}{l} \hat{d}=\Sigma_{k}^{-1} U_{k}^{T} d \\ * \hat{q}=\Sigma_{k}^{-1} U_{k}^{T} q \end{array} $ Match the obtained "semantic" vector representations $\hat{d}$ and $\hat{q}$ using cosine similarity --- ## References