How to know if it's a linear regression problem when working on multi dimensional data? I am working on a supervised learning problem in which my model has to estimate a real value based on a an input vector (of length 10), and I am not sure whether a linear regression problem is applicable to my dataset.
With 2 dimensional data, plotting the data allows us to see if there's a linear relationship between the 2 dimensions. But when the data is highly dimensional (10 dimensions), it's impossible to check if a linear regression model is applicable merely by visualizing the data (which in itself is impossible).
Is there a way (analytical or else) to determine if it's a linear regression problem?
 A: It's good that you are using visualistion.  The data should always be plotted, wherever possible
You can simply extend the same technique that you used for 2 dimensions into 10 dimensions, where you make 10 seperate plots, with the response plotted against each of the 10 features in turn.
Of course this won't identify things like interaction effects, and it will be a good idea to ensure the features are on similar scales (so that nonlinear associations aren't easily masked), but it will go a long way towards identifying any obvious nonlinearities.
There are also a lot of exploratory data analysis techniques that you could use, such as fitting a linear model and then inspecting the residuals for nonlinearity.
Also, don't forget that, just because you might find an obvious non-linear association, you may still be able to use a linear model, by introducing non-linear terms, splines or other transformations (eg log). Linear regression models can be surprisingly flexible.
A: In general it is difficult to identify, whether the problem is linear or non-linear, especially in high-dimensions.
Also there is a Jonhson-Lindenstrauss Lemma :

Let $N \gg 1$. For any $0 < \varepsilon < 1$ and $m$ points in $\mathbb{R}^N$ and
$n > \frac{8 \log m}{\varepsilon^2}$ there exists a linear map $f : \mathbb{R}^N \rightarrow R^n$, such that:
$$
(1 - \varepsilon) \Vert u - v \Vert^2 \leqslant \Vert f(u) - f(v) \Vert^2 \leqslant (1 + \varepsilon) \Vert u - v \Vert^2
$$

So if the dimension is high enough in comparison to the number of points - any problem can be in principle reduced to the linear one, which, however, doesn't mean in practice.
$10$ dimensions is not too much, so maybe it is worth plotting the label against the data for pair of features - to detect pairwise interactions - pairplot from seaborn is a good tool for this.
One may try to perform PCA to 2d and then plot the data with the labels.
