While performing spectral clustering of the original data $\{x_1,...x_n\}$, $ x_i\in \mathbb{R}^{d\times 1}$ (column vectors), into $k$ clusters, we take the first (smallest) $k$ (column) eigenvectors $\{u_1,...u_k\}$ of the Laplacian $L=D-W$, where, $D=$degree matrix, $W=$ weight or affinity matrix, then take the rows $\{y_i:1\leq i \leq n\}$ of $U$ and then perform k-means clustering or a distortion minimizing clustering on these rows. See the screenshot of the algorithm here:

My questions are these:

(1) Let the rows of $U\in \mathbb{R}^{n\times k}$ be $\{y_i:1\leq i \leq n\}$. Then the $k$-dimensional vectors $y_i$'s represent a reduced dimensional interpretation of the original data. But is this lower dimensional representation of the original data any meaningful either geometrically or statistically? By 'geometrically meaningful' I mean: the map $x_i\to y_i$ is locally distance preserving or angle preserving, or at least approximately preserving. By 'statistically meaningful' I mean the same map preserves some statistical properties of the original data, e.g. mean pr covariance, remains the same or approximately the same when we project the data into $\mathbb{R}^{k}$ using the matrix $U$.

(2) Since the distortion minimizing algorithms like k-means that we apply in the final step, assumes that clusters are globular (i.e. "like a ball" or convex from a Euclidean point of view), I wonder what properties do the reduced dimensional representation of the data have that allows us to apply k-means? For example, If we the new data $Y=\{y_1,...y_n\}$, what's the guarantee that it'll have convex clusters?

While performing spectral clustering of the original data $\{x_1,...x_n\}$, $ x_i\in \mathbb{R}^{d\times 1}$ (column vectors), into $k$ clusters, we

Step 1: take the first (smallest) $k$ (column) eigenvectors $\{u_1,...u_k\}$ of the Laplacian $L=D-W$, where, $D=$degree matrix, $W=$ weight or affinity matrix,

Step 2: form the matrix $U\in \mathbb{R}^{n\times k}$ with $u_i$'s as columns,

Step 3: then take the rows $\{y_i:1\leq i \leq n\}$ of $U$ and then perform k-means clustering or a distortion minimizing clustering on these rows. See the screenshot of the algorithm here:

My questions are these:

(1) Let the rows of $U\in \mathbb{R}^{n\times k}$ be $\{y_i:1\leq i \leq n\}$. Then the $k$-dimensional vectors $y_i$'s represent a reduced dimensional interpretation of the original data. But is this lower dimensional representation of the original data any meaningful either geometrically or statistically? By 'geometrically meaningful' I mean: is the map $G: \mathbb{R}^d \to \mathbb{R}^k, G(x_i)= y_i$ locally distance preserving or angle preserving, or at least approximately preserving? By 'statistically meaningful' I mean: if the same map $G$ preserves some statistical properties of the original data, e.g. mean pr covariance, remains the same or approximately the same when we project the data into $\mathbb{R}^{k}$ using the matrix $U$.

(2) Since the distortion minimizing algorithms like k-means that we apply in the final step, assumes that clusters are globular (i.e. "like a ball" or convex from a Euclidean point of view), I wonder what properties do the reduced dimensional representation of the data have that allows us to apply k-means? For example, If we the new data $Y=\{y_1,...y_n\}$, what's the guarantee that it'll have convex clusters?

While performing spectral clustering of the original data $\{x_1,...x_n\}$, $ x_i\in \mathbb{R}^{d\times 1}$ (column vectors), into $k$ clusters, we take the first (smalest) $k$ eigenvectors of the Laplacian $L=D-W$. Here, $D=$degree matrix, $W=$ weight or affinity matrix. Hence we first form an $n \times k$ dimensional representation of the data, with normally $k<d$. i.e. we first take the $k$ eigenvectors with the smallest $k$ eigenvalues, namely, $\{u_1,...u_k\}$ of the Laplacian matrix $L$, and form an $n\times k$ -dimensional data matrix $U$ with $u_i$'s as columns. We then take the rows of $U$ and then perform k-means clustering or a distortion minimizing clustering on these rows. See the screenshot of the algorithm here:

My questions are these:

(1) Let the rows of $U\in \mathbb{R}^{n\times k}$ be $\{y_i:1\leq i \leq n\}$. Then the $k$-dimensional vectors $y_i$'s represent a reduced dimensional interpretation of the original data. But is this lower dimensional representation of the original data any meaningful either geometrically or statistically? By 'geometrically meaningful' I mean: the map $x_i\to y_i$ is locally distance preserving or angle preserving, or at least approximately preserving. By 'statistically meaningful' I mean the same map preserves some statistical properties of the original data, e.g. mean pr covariance, remains the same or approximately the same when we project the data into $\mathbb{R}^{k}$ using the matrix $U$.

(2) Since the distortion minimizing algorithms like k-means that we apply in the final step, assumes that clusters are globular (i.e. "like a ball" or convex from a Euclidean point of view), I wonder what properties do the reduced dimensional representation of the data have that allows us to apply k-means? For example, If we the new data $Y=\{y_1,...y_n\}$, what's the guarantee that it'll have convex clusters?

While performing spectral clustering of the original data $\{x_1,...x_n\}$, $ x_i\in \mathbb{R}^{d\times 1}$ (column vectors), into $k$ clusters, we take the first (smallest) $k$ (column) eigenvectors $\{u_1,...u_k\}$ of the Laplacian $L=D-W$, where, $D=$degree matrix, $W=$ weight or affinity matrix, then take the rows $\{y_i:1\leq i \leq n\}$ of $U$ and then perform k-means clustering or a distortion minimizing clustering on these rows. See the screenshot of the algorithm here:

My questions are these:

(1) Let the rows of $U\in \mathbb{R}^{n\times k}$ be $\{y_i:1\leq i \leq n\}$. Then the $k$-dimensional vectors $y_i$'s represent a reduced dimensional interpretation of the original data. But is this lower dimensional representation of the original data any meaningful either geometrically or statistically? By 'geometrically meaningful' I mean: the map $x_i\to y_i$ is locally distance preserving or angle preserving, or at least approximately preserving. By 'statistically meaningful' I mean the same map preserves some statistical properties of the original data, e.g. mean pr covariance, remains the same or approximately the same when we project the data into $\mathbb{R}^{k}$ using the matrix $U$.

(2) Since the distortion minimizing algorithms like k-means that we apply in the final step, assumes that clusters are globular (i.e. "like a ball" or convex from a Euclidean point of view), I wonder what properties do the reduced dimensional representation of the data have that allows us to apply k-means? For example, If we the new data $Y=\{y_1,...y_n\}$, what's the guarantee that it'll have convex clusters?