TLDR: given two tensors $t_1$ and $t_2$, both with shape $(c,h,w)$ what metric should be use to measure their distance?

More Info: I'm working on a project in which I'm trying to distinguish between an anomalous sample $A$ (specifically from MNIST) and a "regular" sample $R$. The solution I chose is to consider the feature maps that are given by ResNet and use kNN. More specifically:

• I embed the entire CIFAR10_TRAIN data to achieve a dataset that consists of activations with dimension $(N,c,h,w)$ where $N$ is the size of CIFAR_TRAIN
• For $2$ new test samples $t_C$ and $t_M$ from CIFAR10_TEST and MNIST_TEST respectively (both with shape $(c,h,w)$) I embed them both to receive the activations (same as for the training data)
• (!) I find the k-Nearest-Neighbours of $t_C$ and $t_M$ w.r.t the embedding of the training data
• I calculate the mean distance between the $k$ neighbors
• Given some predefined threshold, I classify $t_C$ and $t_M$ as regular or anomalous, hoping that the distance for $t_M$ would be higher, as it represents O.O.D sample.

Notice that in (!) I need some distance measure, but this is not trivial as these are tensors, not vectors.

What I've Tried: a trivial solution is to flatten the tensor to have shape $(c\cdot h\cdot w)$ and then use basic $\ell_2$, but the results turned out pretty bad. (could not distinguish regular vs anomalous in this case). Hence: Is there a better way of measuring this distance?

TLDR: given two tensors $t_1$ and $t_2$, both with shape $(c,h,w),$ how shall the distance between them be measured?

More Info: I'm working on a project in which I'm trying to distinguish between an anomalous sample (specifically from MNIST) and a "regular" sample (specifically from CIFAR10). The solution I chose is to consider the feature maps that are given by ResNet and use kNN. More specifically:

• I embed the entire CIFAR10_TRAIN data to achieve a dataset that consists of activations with dimension $(N,c,h,w)$ where $N$ is the size of CIFAR_TRAIN
• I embed $2$ new test samples $t_C$ and $t_M$ from CIFAR10_TEST and MNIST_TEST respectively (both with shape $(c,h,w)$), same as I did with the training data.
• (!) I find the k-Nearest-Neighbours of $t_C$ and $t_M$ w.r.t the embedding of the training data
• I calculate the mean distance between the $k$ neighbors
• Given some predefined threshold, I classify $t_C$ and $t_M$ as regular or anomalous, hoping that the distance for $t_M$ would be higher, as it represents O.O.D sample.

Notice that in (!) I need some distance measure, but this is not trivial as these are tensors, not vectors.

What I've Tried: a trivial solution is to flatten the tensor to have shape $(c\cdot h\cdot w)$ and then use basic $\ell_2$, but the results turned out pretty bad. (could not distinguish regular vs anomalous in this case). Hence: Is there a better way of measuring this distance?