Linear Regression for Noisy Data I have noisy dataset collected from a source and I am planning to fit a regression to this dataset.
The dataset has Y and X1 variables (both continuous between (-1, 1)) and I plotted a scatter plot to examine if it is valid to use regression or not. To me a there is no clear relationship between these two variables even the correlation is greater than 0.5. There is a sign of clustering though.  If I split these two clusters and fit a regression for each cluster individually, the relationship will be weak.
Is there any hope that we can use regression analysis to this problem?
Update
Regarding the dataset. I have a black box model (API) the receives a text as an input and gives a sentiment score between [-1, 1]. I've collected a sample of texts X, and these texts are labeled by humans already in terms of a correct sentiment score Y. Once I received the results from the black box models, I plotted a scatter plot between the results from this API and a ground truth score labeled by humans. This is what the graph below is showing.
My approach is trying to examine the accuracy of this API and also I am trying to understand the underlying relationship and how does it treat different datapoint. My assumption is if this API is really accurate, we could see a somehow a relationship in the scatter plot.

 A: Linear regression would not be a sensible approach for the data in that paper, since the relationship between X and Y is does not linear (conditional on knowing the clustering, there doesnt seem to be any relationship between X and Y at all, and any 'linearity' you find is going to be a spurious result of marginalising over the cluster allocations). You should break the observations into clusters first, and fit a separate model within each cluster.
This really looks like a problem of omitted variables. There is perhaps some variable X2 (which you havent measured) which is separating the observations into clusters. If you fit a model which included this X2 and added in the interaction terms (essentially leading a multi-level model with random intercepts and perhaps random slopes, see https://en.wikipedia.org/wiki/Multilevel_model) then its coefficient would be highly significant. If you cant measure X2 directly then doing some kind of clustered regression where X2 is the cluster index of each observation would perhaps be sensible. This could be done either as a preprocessing step (i.e. you first cluster the data, take the cluster allocations as fixed, and then run a regression) or you could do some kind of full Bayesian analysis where you simultaneously learn the clusters and the regression model within each cluster and have full uncertainty quantification over everything.
A: After your update, I can say by eye that your model seems to separate low and high Y well. The Ys are clustered. The fact that you observe that also the Xs are clustered in a similar way means that they are capturing something related to the Ys.
You can estimate the predictive power by the mean squared error between the X and Y.
Also, you can binarize the two variables setting a threshold equal to 0 and estimate the accuracy.
It depends on what you are trying to predict. If you need to predict the continuous values of the Ys then probably you need to optimise the black box model generating the Xs.
You can find which text have the largest error and try to figure out if they have some property in common, that the current X is not capturing.
