Package: DeCovarT 2.2.3

Bivariate toy model: when covariance hurts mean-only deconvolution | Generative design | How to run it | High-dimensional hybrid scenario and deconvolution | Gene and cell-type design | $$\underbrace{\mathcal{G}{12}}{30\text{ genes}}\ \cup\underbrace{\mathcal{G}{3}}{10\text{ genes}}\ \cup\underbrace{\mathcal{G}{\mathrm{eq}}}{10\text{ genes}},\qquad\boldsymbol{\Omega}_j | Deconvolution on imbalanced mixtures | Related literature | References

Notation | Matrix calculus reminders | First-order matrix calculus | $$\begin{aligned}\frac{\partial\det(\boldsymbol{A})}{\partial p}&=\det(\boldsymbol{A}),\mathrm{tr}!\left(\boldsymbol{A}^{-1}\frac{\partial\boldsymbol{A}}{\partial p}\right),\\frac{\partial(\boldsymbol{U}\boldsymbol{A}\boldsymbol{V})}{\partial p}&=\boldsymbol{U}\frac{\partial\boldsymbol{A}}{\partial p}\boldsymbol{V},\\frac{\partial\boldsymbol{A}^{-1}}{\partial p}&= | $$\frac | \mathrm | $$\frac{\partial}{\partial p}\bigl((\boldsymbol{y}-\boldsymbol{x}p)^{\top}\boldsymbol{\Theta}(\boldsymbol{y}-\boldsymbol{x}p)\bigr) | Second-order matrix calculus | $$\frac | \boldsymbol | $$\frac | Inner products in the score | Unconstrained score and Hessian | $$\ell_{\boldsymbol{y}\mid\boldsymbol{\zeta}}(\boldsymbol{p}) | \log\det\boldsymbol{\Theta}(\boldsymbol{p}) | What “unconstrained” means here | Unconstrained gradient | Unconstrained Hessian | $$\begin | \underbrace | \boldsymbol | $$\begin | \underbrace | \boldsymbol | Constrained optimisation on the simplex | Log-sum-exp and softmax | $$s_ | \frac | e^{z_{i}-\operatorname{LSE}(\boldsymbol{z})},\qquad\operatorname{LSE}(\boldsymbol{z}) | $$H_ | s_{i}\bigl[(\delta_{ij}-s_{j})(\delta_{ik}-s_{k}) | Additive log-ratio map | $$p_ | \frac | \frac | ALR is a $C^{2}$ diffeomorphism | Efficient derivatives of $\boldsymbol | $$\frac | Change of variables: ambient $\boldsymbol | Constrained score and Hessian | $$\frac | Implementation map | Expected Fisher information and Wald inference | Unconstrained expected Fisher information | $$I(\boldsymbol | Observed versus expected information | Constrained case: ALR delta method | $$I_ | Numerical speed-ups and solver safeguards | Cache a Cholesky factorisation of $\boldsymbol{\Sigma}(\boldsymbol{p})$ | Guard the box-constrained L-BFGS-B path | marqLevAlg Hessian sign under minimize = FALSE | marqLevAlg(minimize = FALSE) does not flip hess | References