I've began to study the relationship between calibrations and holonomy, mainly through D.D. Joyce's Riemannian Holonomy Groups and Calibrated Geometry and partly through internet material. Pretty much everyone explains this relationship by the holonomy principle: if $H=\text{Hol}_x$ and $\varp...
The fundamental lemma of calibrated geometry states that calibrated submanifolds are absolutely volume minimising in their homology class. In the proof, homology equivalent is used synonymously with cobordant. With respect to which homology theory are they minimising? Are they volume minimising w...
I am reading the paper "Unsupervised Learning of Depth and Ego-Motion from Video" (https://people.eecs.berkeley.edu/~tinghuiz/projects/SfMLearner/cvpr17_sfm_final.pdf) Equation 2 of this paper is as follows: $$p_s \approx K\hat{T}_{t\rightarrow s}\hat{D}(p_t)K^{-1}p_t$$ It projects a pixel in t...
In the original Platt Scaling paper [1], the target probabilities used to calibrate a model's non-probabilistic outputs are found to be: $$t_+ = \frac{N_+ + 1}{N_+ + 2} \qquad t_- = \frac{1}{N_- + 2} $$ How are these, as the paper says, max a posteriori estimates? (What are the prior, likeliho...
$u: \Omega \subset \mathbb R^2 \rightarrow \mathbb R$ is a $C^2$ function. Graph of $u$ is $$ G_u=\{(x,y,u(x,y)) : (x,y)\in \Omega\} $$ And the upward pointing unit normal is $N$. $\omega$ is the two-form on $\Omega\times \mathbb R$ given by that for $X,Y\in \mathbb R^3$ $$ \omega(X,Y) = \det (X...
Rename calibrations to calibration-geometry. The tag calibrations was created in June 2019. This is the question where the tag was created: Calibrations vs. Riemannian holonomy. From this question it seems that the intention was to created a tag for calibrated geometry. As the tag name can be in...
Rename calibrations to calibrated-geometry. (Also a name such as (calibrations-differential-geometry) - or something similar - would help with clarifying the content of the tag. However, that seems to be unnecessarily long.) The tag calibrations was created in June 2019. This is the question whe...
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