Merge branch 'main' of github.com:zahl-/Interview-Task

README.md

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+# Research Software Engineer Pre-Interview Task
+
+This repo contains the Non-negative Matrix Factorization (NNMF) library and the Latex files for the pre-interview task.
+
+## The NNMF library
+Let $A$ be a nonnegative $m \times n$ matrix where the $n$ column vectors are viewed at data points with $m$ features.
+The goal is to factor $A$ into two nonnegative matrices $A \approx WH$ where the dimensions of $W$ and $H$ are 
+$m \times k$ and $k \times n$. $W$ is the matrix of centroids, an $H$ is the coefficient matrix.
+
+Our implementation follows three steps:
+1. Initialization - we initialize $W$ and $H$ via $k$-means clustering. $W$ is initialized to the matrix consisting of the $k$ centroids, and $H$ is initialized to the indicator matrix which assigns each vector to a cluster
+2. Update - we update $W$ and $H$ by applying non-negative least squares (NLS) in an alternating manner. That is, we update the rows of $W$ by applying NLS to $H$ and $A$, then update the columns of $H$ by applying NLS to $W$ and $A$.
+3. Evalutation - we evaluate the solution with the Frobenious norm $\||A - WH\||_F$.
+
+#### A note on notation
+In this implementation we rely on ```sklearn.cluster.KMeans``` for our initialization. This package assumes the $m \times n$
+matrix consists of $m$ data points with $n$ features. This is the transpose of the setup given in the pre-interview task.
+As a result, our implementation formulates the problem as $A^T \approx H^T W^T$.
+
+### Documentation
+The library is contained in ```code/nnmf.py``` and consists of the four functions defined here.
+| Function   | Input                                                                                                                                               | Output                                        | Description                                                                                                                            |
+|------------|-----------------------------------------------------------------------------------------------------------------------------------------------------|-----------------------------------------------|----------------------------------------------------------------------------------------------------------------------------------------|
+| initialize | <code>A: numpy.ndarray</code><br><br> <code>k: int</code>                                                                                               | <code>(numpy.ndarray, numpy.ndarray)</code>   | Returns the initial factorization.                                                                                                     |
+| update     | <code>A: numpy.ndarray</code><br><br> <code>H: numpy.ndarray</code><br><br> <code>W: numpy.ndarray</code>                                                   | <code> (numpy.ndarray, numpy.ndarray) </code> | Performs one step of the alternating NLS update and return <code>(H, W)</code>                                                                                     |
+| fnorm      | <code>m: numpy.ndarray</code>                                                                                                                       | <code>float</code>                            | Returns the Frobenious norm of a matrix.                                                                                               |
+| loss       | <code> A: numpy.ndarray </code><br><br> <code> H: numpy.ndarray </code><br><br> <code> W: numpy.ndarray </code>                                             | <code>float</code>                            | Returns <code>fnorm(A - H@W)</code>.                                                                                                   |
+| nnmf       | <code>A: numpy.ndarray</code><br><br> <code>k: int</code><br><br> <code>max_iter: Optional[int] = 1000</code><br><br> <code>tol: Optional[float] = 0.001</code> | <code> (numpy.ndarray, numpy.ndarray) </code> | Performs the NNMS algorithm. Terminate after <code>max_iter</code> iterations or after achieving an error tolerance of <code>tol</code>. Returns <code>(H, W)</code> |