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On page 576 in Bishop's PRML, it is stated that

$$ (\mathbf{W}_{ML}^T\mathbf{W}_{ML})^{-1}\mathbf{W}^T_{ML}(\mathbf{x} - \mathbf{\bar{x}}) $$

represents an orthogonal projection of the data point $\mathbf{x}$ onto the latent space.

$\mathbf{W}$ is a $D\times M$ matrix. $\mathbf{x}$ is $D\times 1$. The latent space is $M$-dimensional.

What does it mean that the expression represents an orthogonal projection (and how do we know that it is one) onto the latent space and why is it important?

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This question arises in the context of principal component analysis. To keep our notation simple, let's assume that our dataset is centred at the origin, i.e. $\mathbf {\bar x }= 0$. The goal is to approximate a datapoint $\mathbf x \in \mathbb R^n$ as closely as possible using a point chosen from the $d$-dimensional subspace spanned by the columns of an $n \times d$ matrix $\mathbf W$. In other words, we wish to find $$ \mathbf x_\star := \mathbf W\mathbf z_\star,$$

where $$ \mathbf z_{\star}:= {\rm argmin}_{\mathbf z \in \mathbb R^d}\| \mathbf x - \mathbf W \mathbf z\|^2.$$

[This $\mathbf x_\star$ is called the "orthogonal projection" of $\mathbf x$ onto the hyperplane spanned by the columns of $\mathbf W$, because, if computed correctly, $\mathbf x_\star$ will be orthogonal to the displacement vector from $\mathbf x_\star$ to $\mathbf x$. Geometrically, this is quite intuitive.]

Let's go ahead and compute this approximation. First, let's find $\mathbf z_\star$, by differentiation: $$ \mathbf 0 = \left( \frac{\partial}{\partial \mathbf z} \| \mathbf x - \mathbf W \mathbf z\|^2 \right)\vert_{{\mathbf z = \mathbf z_\star}} = -2\mathbf W^T(\mathbf x - \mathbf W\mathbf z_\star) \implies \mathbf z_\star = (\mathbf W^T \mathbf W)^{-1}\mathbf W^T \mathbf x.$$

In machine learning, this $\mathbf z_{\star}$ is the latent vector for this datapoint, and corresponds to the expression in your question (assuming $\mathbf {\bar x} = 0$). The approximation $\mathbf x_\star$ is then given by $\mathbf x_\star = \mathbf W \mathbf z_\star$.

[Just for fun, let's verify that $\mathbf x_\star$ and $\mathbf x - \mathbf x - \mathbf x_\star$ are orthogonal, justifying the phrase "orthogonal projection": \begin{align} \mathbf x_\star . (\mathbf x - \mathbf x_\star) &= \mathbf x\mathbf W(\mathbf W^T\mathbf W)^{-1}\mathbf W^T\left( \mathbf x-\mathbf W(\mathbf W^T\mathbf W)^{-1}\mathbf W^T \mathbf x\right) \\ &= \mathbf x\mathbf W(\mathbf W^T\mathbf W)^{-1}\mathbf W^T \mathbf x- \mathbf x\mathbf W(\mathbf W^T\mathbf W)^{-1}\mathbf W^T \mathbf x \\ &= 0. \end{align} ]

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I don't know what machine learning or a latent space is, but I understand orthogonal projections. x is a data point and you want to find the closest point y in the latent space. This is done by orthogonal projection. Geometrically, you can write x as a linear combination of ($M$+1) orthogonal vectors, $M$ of which are in the latent space. The remaining vector, call it z is the orthogonal complement of x.

Note $\mid\mid$z$\mid\mid$ is the distance from x to the latent space.

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