What is an SVM, anyway?
I think the answer for most purposes is “the solution to the following optimization problem”:
$$
\begin{split}
\operatorname*{arg\,min}_{f \in \mathcal H} \frac{1}{n} \sum_{i=1}^n \ell_\mathit{hinge}(f(x_i), y_i) \, + \lambda \lVert f \rVert_{\mathcal H}^2
\\ \ell_\mathit{hinge}(t, y) = \max(0, 1 - t y)
,\end{split}
\tag{SVM}
$$
where $\mathcal H$ is a reproducing kernel Hilbert space, $y$ is a label in $\{-1, 1\}$, and $t = f(x) \in \mathbb R$ is a “decision value”; our final prediction will be $\operatorname{sign}(t)$. In the simplest case, $\mathcal H$ could be the space of affine functions $f(x) = w \cdot x + b$, and $\lVert f \rVert_{\mathcal H}^2 = \lVert w \rVert^2 + b^2$. (Handling of the offset $b$ varies depending on exactly what you’re doing, but that’s not important for our purposes.)
In the ‘90s through the early ‘10s, there was a lot of work on solving this particular optimization problem in various smart ways, and indeed that’s what LIBSVM / LIBLINEAR / SVMlight / ThunderSVM / ... do. But I don’t think that any of these particular algorithms are fundamental to “being an SVM,” really.
Now, how do we train a deep network? Well, we try to solve something like, say,
$$
\begin{split}
\operatorname*{arg\,min}_{f \in \mathcal F} \frac1n \sum_{i=1}^n \ell_\mathit{CE}(f(x_i), y) + R(f)
\\
\ell_\mathit{CE}(p, y) = - y \log(p) - (1-y) \log(1 - p)
,\end{split}
\tag{$\star$}
$$
where now $\mathcal F$ is the set of deep nets we consider, which output probabilities $p = f(x) \in [0, 1]$. The explicit regularizer $R(f)$ might be an L2 penalty on the weights in the network, or we might just use $R(f) = 0$. Although we could solve (SVM) up to machine precision if we really wanted, we usually can’t do that for $(\star)$ when $\mathcal F$ is more than one layer; instead we use stochastic gradient descent to attempt at an approximate solution.
If we take $\mathcal F$ as a reproducing kernel Hilbert space and $R(f) = \lambda \lVert f \rVert_{\mathcal F}^2$, then $(\star)$ becomes very similar to (SVM), just with cross-entropy loss instead of hinge loss: this is also called kernel logistic regression. My understanding is that the reason SVMs took off in a way kernel logistic regression didn’t is largely due to a slight computational advantage of the former (more amenable to these fancy algorithms), and/or historical accident; there isn’t really a huge difference between the two as a whole, as far as I know. (There is sometimes a big difference between an SVM with a fancy kernel and a plain linear logistic regression, but that’s comparing apples to oranges.)
So, what does a deep network using an SVM to classify look like? Well, that could mean some other things, but I think the most natural interpretation is just using $\ell_\mathit{hinge}$ in $(\star)$.
One minor issue is that $\ell_\mathit{hinge}$ isn’t differentiable at $\hat y = y$; we could instead use $\ell_\mathit{hinge}^2$, if we want. (Doing this in (SVM) is sometimes called “L2-SVM” or similar names.) Or we can just ignore the non-differentiability; the ReLU activation isn’t differentiable at 0 either, and this usually doesn’t matter. This can be justified via subgradients, although note that the correctness here is actually quite subtle when dealing with deep networks.
An ICML workshop paper – Tang, Deep Learning using Linear Support Vector Machines, ICML 2013 workshop Challenges in Representation Learning – found using $\ell_\mathit{hinge}^2$ gave small but consistent improvements over $\ell_\mathit{CE}$ on the problems they considered. I’m sure others have tried (squared) hinge loss since in deep networks, but it certainly hasn’t taken off widely.
(You have to modify both $\ell_\mathit{CE}$ as I’ve written it and $\ell_\mathit{hinge}$ to support multi-class classification, but in the one-vs-rest scheme used by Tang, both are easy to do.)
Another thing that’s sometimes done is to train CNNs in the typical way, but then take the output of a late layer as "features" and train a separate SVM on that. This was common in early days of transfer learning with deep features, but is I think less common now.
Something like this is also done sometimes in other contexts, e.g. in meta-learning by Lee et al., Meta-Learning with Differentiable Convex Optimization, CVPR 2019, who actually solved (SVM) on deep network features and backpropped through the whole thing. (They didn't, but you can even do this with a nonlinear kernel in $\mathcal H$; this is also done in some other "deep kernels" contexts.) It’s a very cool approach – one that I've also worked on – and in certain domains this type of approach makes a ton of sense, but there are some pitfalls, and I don’t think it’s very applicable to a typical "plain classification" problem.
Best Answer
Neural networks in computer vision usually operate with inputs of shape
(m, n)— images—so there is nothing unusual about that.I would avoid using fully connected layers in this case and go for a fully-convolutional network. Have a look at U-Net, which has become quite a standard architecture for image-to-image tasks: https://arxiv.org/abs/1505.04597
Feeding it additional data of could be tricky, but fusing them in as additional channels at the bottom part of the U shape could work. Proper scaling might be necessary.