Finally, if you've gotten this far: this work builds on our earlier work with @congma.bsky.social using ~complex analysis~ (conformal mapping) to model spatial variation in ST data: www.sciencedirect.com/science/arti...

Check out the paper for more details and neat biological applications! For example, modeling spatial gradients → much more accurate SVG identification

Thanks to great collaborators: Brian @hrksrkr.bsky.social Kohei @congma.bsky.social Sereno @braphael.bsky.social

GASTON algorithm: parametrize functions h,d with neural nets and learn from data!

Fun back-story: @braphael.bsky.social and I derived most of this model at the bar near an NCI workshop 🥂😅

Model implicitly accounts for sparsity by “pooling” measurements across locations (x,y) with equal isodepth d(x,y).

These locations look like contours of equal height on an elevation map, hence the “topographic map” analogy.

We prove that f(x,y) = h(d(x,y)), i.e. gene expression f(x,y) is function of a *single* spatial coordinate d(x,y) rather than 2 spatial coordinates x,y

-> Spatial dimensionality reduction! 🚀

We call d(x,y) the "isodepth" - it characterizes spatial gradients ▽f_g

We handle sparsity w/ two assumptions:

(1) genes have *shared* gradient directions, i.e. each gradient ▽f_g(x,y) is proportional to shared vector field v(x,y)
(Equivalent to Jacobian of f being rank-1 everywhere)

(2) vector field v has no “curl”, so v=▽d is gradient of "spatial potential" d

You can view ST data as samples of function f: R^2 → R^G, where f(x,y) is (high-dim) gene expression vector at location (x,y).

Spatial gradients are gradients ▽f_g of each component (gene)

Unfortunately, large data sparsity means naive estimation of gradient ▽f_g is very noisy 😱