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Colah, your posts are really inspiring and thoughtful. Your thoughts about visualising the space of representations by looking at the properties of pairwise di

by xtacy 12y ago

Colah, your posts are really inspiring and thoughtful. Your thoughts about visualising the space of representations by looking at the properties of pairwise distance matrix is quite illuminating. It might be a nice empirical way to get a glimpse of the model complexity: If "simpler" models cluster close to more complex models, the simpler models are more desirable. I wonder if all over-fitted models cluster in one region in the meta-SNE space, or do they show up as noise? Keep up the great posts!

colah3 12y ago
Thanks, xtacy! > If "simpler" models cluster close to more complex models, the simpler models are more desirable. Well, it would suggest you aren't winning very much for your more complex model, at the very least. > I wonder if all over-fitted models cluster in one region in the meta-SNE space, or do they show up as noise? This corresponds to an empirical question: do models overfit in the same way, or different ways? One small experiment I did, which might offer some intuition here, was training lots of extremely small networks on MNIST, with hidden layers of only 1, 2 or 5 neurons. What do they look like in meta-SNE? Well, it turns out that when you only have a very small number of neurons, they latch on to random useful features! These randomly selected features don't tend to be the same, so you end up with the models horribly disagreeing on what is similar and what is different. As you increase the number of neurons, the space of features they look at, if not the features of individual neurons, becomes similar across models. And so the models agree more, and cluster more tightly. ... Another fun idea for using meta-SNE is ensemble models. We know that training a bunch of models and then averaging their results (ensembling) can improve results a lot. When is this helpful? My guess is that the farther apart compatibly good models are in meta-SNE space, the more ensembling will help, because they've learned different things.
- xtacy 12y ago
  Ensemble (and also boosted) models: Very nice idea. I like the takeaway that meta-SNE idea is powerful to compare the space of models by through the lens of pairwise distances as a proxy for the distance metric. Are distances the defining property for a vector space R^d? Could you have used some other quantity instead of pairwise distances?
  - colah3 12y ago
    There "the defining property" if you want to mod out isometries. :) They're nice, because they encode the geometry of the data. You could very reasonably try things like cosine distance. And I did some experiments, to good results, with sqrt(d(x,y)), to emphasize really close together data points as special. But these don't feel as motivated. Hm. It might also be interesting to try with the p_ij values from t-SNE, which model the topology of the data. Then you'd really be getting meta. :)
    - xtacy 12y ago
      Interesting. IIUC, what you're implying is that defining a metric defines the topology and they're equivalent. Isn't p_ij in t-SNE also derived from the distances themselves, where p_ij ~ student_t(d_ij, degrees_of_freedom) (I forget how the d.o.f. is actually computed in t-SNE.) Which leads me to one way this distance based approach might be limited: It models similarities using distances, which are symmetric. If similarities aren't symmetric, then this visualisation could hide some information. For example: The specific entity "BMW car" is more similar to the more general entity "car" than the entity "car" is to "BMW car." It seems this asymmetry could capture things (such as the generality of concepts), not reflected in metric spaces (on first thought).

by xtacy 12y ago

colah3 12y ago

Thanks, xtacy! > If "simpler" models cluster close to more complex models, the simpler models are more desirable. Well, it would suggest you aren't winning very much for your more complex model, at the very least. > I wonder if all over-fitted models cluster in one region in the meta-SNE space, or do they show up as noise? This corresponds to an empirical question: do models overfit in the same way, or different ways? One small experiment I did, which might offer some intuition here, was training lots of extremely small networks on MNIST, with hidden layers of only 1, 2 or 5 neurons. What do they look like in meta-SNE? Well, it turns out that when you only have a very small number of neurons, they latch on to random useful features! These randomly selected features don't tend to be the same, so you end up with the models horribly disagreeing on what is similar and what is different. As you increase the number of neurons, the space of features they look at, if not the features of individual neurons, becomes similar across models. And so the models agree more, and cluster more tightly. ... Another fun idea for using meta-SNE is ensemble models. We know that training a bunch of models and then averaging their results (ensembling) can improve results a lot. When is this helpful? My guess is that the farther apart compatibly good models are in meta-SNE space, the more ensembling will help, because they've learned different things.

xtacy 12y ago
Ensemble (and also boosted) models: Very nice idea. I like the takeaway that meta-SNE idea is powerful to compare the space of models by through the lens of pairwise distances as a proxy for the distance metric. Are distances the defining property for a vector space R^d? Could you have used some other quantity instead of pairwise distances?
- colah3 12y ago
  There "the defining property" if you want to mod out isometries. :) They're nice, because they encode the geometry of the data. You could very reasonably try things like cosine distance. And I did some experiments, to good results, with sqrt(d(x,y)), to emphasize really close together data points as special. But these don't feel as motivated. Hm. It might also be interesting to try with the p_ij values from t-SNE, which model the topology of the data. Then you'd really be getting meta. :)
  - xtacy 12y ago
    Interesting. IIUC, what you're implying is that defining a metric defines the topology and they're equivalent. Isn't p_ij in t-SNE also derived from the distances themselves, where p_ij ~ student_t(d_ij, degrees_of_freedom) (I forget how the d.o.f. is actually computed in t-SNE.) Which leads me to one way this distance based approach might be limited: It models similarities using distances, which are symmetric. If similarities aren't symmetric, then this visualisation could hide some information. For example: The specific entity "BMW car" is more similar to the more general entity "car" than the entity "car" is to "BMW car." It seems this asymmetry could capture things (such as the generality of concepts), not reflected in metric spaces (on first thought).