## Expanding Inverse Functions

The problem of inverting the implicit function $$y=f(x)$$ in the form of power-series $$x = a + \sum_{k=1}^{\infty} b_k (y-f(a))^k$$ around a point of interest $x=a$, has a long history. Lagrange obtained a theoretical inversion formula [1]https://mathworld.wolfram.com/LagrangeInversionTheorem.html, yet efficient implementations are relatively recent [2]Brent, Richard P., and Hsiang T. Kung. „Fast algorithms for manipulating formal …

## Iron logic defeats strawman

The straw man rhetorical technique is widely used: from scientific reviews (academia), through job interviews (industry) to the political discourse (international relations). The idea is simple: you distort arguments or opposing views to easier refute them. Not only is this trick popular, but proven successful statistically. The best weapon against it is logically strict reasoning. …

## Can Russia defend its economy from sanctions?

While western leaders are proud of imposing largest sanctions ever, it seems that Russia can withstand the crisis longer than some expect. The following two points, underrepresented in popular media, support this claim: Historical AnalogyRussia’s went through 2014 financial crisis, where the currency significantly lost its value (from mid 30-s to around 80 for one …

## On Subgaussian Concentration of Missing Mass

The problem of missing mass goes back to the cryptographic work of Good and Turing during WWII, but has been also studied in the context of linguistic, ecology, and by probability theoreticians. The missing mass is defined as the weight of elements not observed in a sample, due to pure chance: Such an event and …

## Approximating Tails of Beta Distribution

Beta distribution is ubiquitous in statistics, but particularly popular in real-world modeling. The beta-binomial model is perhaps the most known example, given the recent interest in Bayesian inference. But it was in use nearly 50 years ago, for example in toxicology. Unfortunately, computing probabilities from the density depends on intractable incomplete beta integrals. This creates …

## Improving State-of-Art on Sparse Random Projections

Random projections are widely used to reduce data dimension in various analyses. Provable guarantees were developed first in the important result of Johnson and Lindenstrauss on Lipschitz maps, but more recently there has been a lot of follow-up work in the context of machine-learning. Particularly attractive are sparse random projections, which share similar guarantees as …