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In machine learning, the kernel embedding of distributions (also called the kernel mean or mean map) comprises a class of nonparametric methods in which a probability distribution is represented as an element of a reproducing kernel Hilbert space (RKHS). A generalization of the individual data-point feature mapping done in classical kernel methods, the embedding of distributions into infinite-dimensional feature spaces can preserve all of the statistical features of arbitrary distributions, while allowing one to compare and manipulate distributions using Hilbert space operations such as inner products, distances, projections, linear transformations, and spectral analysis. This learning framework is very general and can be applied to distributions over any space ${\displaystyle \Omega }$ on which a sensible kernel function (measuring similarity between elements of ${\displaystyle \Omega }$) may be defined. For example, various kernels have been proposed for learning from data which are: vectors in ${\displaystyle \mathbb {R} ^{d}}$, discrete classes/categories, strings, graphs/networks, images, time series, manifolds, dynamical systems, and other structured objects. The theory behind kernel embeddings of distributions has been primarily developed by Alex Smola, Le Song , Arthur Gretton, and Bernhard Schölkopf. A review of recent works on kernel embedding of distributions can be found in.

The analysis of distributions is fundamental in machine learning and statistics, and many algorithms in these fields rely on information theoretic approaches such as entropy, mutual information, or Kullback–Leibler divergence. However, to estimate these quantities, one must first either perform density estimation, or employ sophisticated space-partitioning/bias-correction strategies which are typically infeasible for high-dimensional data. Commonly, methods for modeling complex distributions rely on parametric assumptions that may be unfounded or computationally challenging (e.g. Gaussian mixture models), while nonparametric methods like kernel density estimation (Note: the smoothing kernels in this context have a different interpretation than the kernels discussed here) or characteristic function representation (via the Fourier transform of the distribution) break down in high-dimensional settings.

Methods based on the kernel embedding of distributions sidestep these problems and also possess the following advantages:

1. Data may be modeled without restrictive assumptions about the form of the distributions and relationships between variables
2. Intermediate density estimation is not needed
3. Practitioners may specify the properties of a distribution most relevant for their problem (incorporating prior knowledge via choice of the kernel)
4. If a characteristic kernel is used, then the embedding can uniquely preserve all information about a distribution, while thanks to the kernel trick, computations on the potentially infinite-dimensional RKHS can be implemented in practice as simple Gram matrix operations
5. Dimensionality-independent rates of convergence for the empirical kernel mean (estimated using samples from the distribution) to the kernel embedding of the true underlying distribution can be proven.
6. Learning algorithms based on this framework exhibit good generalization ability and finite sample convergence, while often being simpler and more effective than information theoretic methods

Thus, learning via the kernel embedding of distributions offers a principled drop-in replacement for information theoretic approaches and is a framework which not only subsumes many popular methods in machine learning and statistics as special cases, but also can lead to entirely new learning algorithms.

🔗 United States/U.S. Government 🔗 United States 🔗 Philosophy 🔗 Politics 🔗 Philosophy/Social and political philosophy 🔗 Philosophy/Contemporary philosophy 🔗 United States Public Policy 🔗 United States/U.S. Public Policy

The Overton window is the range of policies politically acceptable to the mainstream population at a given time. It is also known as the window of discourse. The term is named after Joseph P. Overton, who stated that an idea's political viability depends mainly on whether it falls within this range, rather than on politicians' individual preferences. According to Overton, the window frames the range of policies that a politician can recommend without appearing too extreme to gain or keep public office given the climate of public opinion at that time.

🔗 Internet culture

Zalgo text, also known as cursed text due to the nature of its use, is digital text that has been modified with numerous combining characters, Unicode symbols used to add diacritics above or below letters, to appear frightening or glitchy.

Named for a 2004 Internet creepypasta story that ascribes it to the influence of an eldritch deity, Zalgo text has become a significant component of many Internet memes, particularly in the "surreal meme" culture. The formatting of Zalgo text also allows it to be used to halt or impair certain computer functions, whether intentionally or not.

🔗 Mathematics

A sequence of six 9's occurs in the decimal representation of the number pi (π), starting at the 762nd decimal place. It has become famous because of the mathematical coincidence and because of the idea that one could memorize the digits of π up to that point, recite them and end with "nine nine nine nine nine nine and so on", which seems to suggest that π is rational. The earliest known mention of this idea occurs in Douglas Hofstadter's 1985 book Metamagical Themas, where Hofstadter states

I myself once learned 380 digits of π, when I was a crazy high-school kid. My never-attained ambition was to reach the spot, 762 digits out in the decimal expansion, where it goes "999999", so that I could recite it out loud, come to those six 9's, and then impishly say, "and so on!"

This sequence of six nines is sometimes called the "Feynman point", after physicist Richard Feynman, who allegedly stated this same idea in a lecture. It is not clear when, or even if, Feynman made such a statement, however; it is not mentioned in published biographies or in his autobiographies, and is unknown to his biographer, James Gleick.

🔗 Philosophy 🔗 Philosophy/Contemporary philosophy 🔗 Philosophy/Ethics

Master–slave morality (German: Herren- und Sklavenmoral) is a central theme of Friedrich Nietzsche's works, particularly in the first essay of his book, On the Genealogy of Morality. Nietzsche argued that there were two fundamental types of morality: "master morality" and "slave morality". Master morality values pride and power, while slave morality values kindness, empathy, and sympathy. Master morality judges actions as good or bad (e.g. the classical virtues of the noble man versus the vices of the rabble), unlike slave morality, which judges by a scale of good or evil intentions (e. g. Christian virtues and vices, Kantian deontology).

For Nietzsche, a morality is inseparable from the culture which values it, meaning that each culture's language, codes, practices, narratives, and institutions are informed by the struggle between these two moral structures.

🔗 United States 🔗 Disaster management 🔗 Crime 🔗 Death 🔗 Lists 🔗 Politics 🔗 Politics/American politics 🔗 Years 🔗 United States/U.S. history 🔗 Current events 🔗 Politics/Gun politics

This is a list of shootings in the United States that have occurred in 2022. Mass shootings are incidents involving several victims of firearm-related violence. The precise inclusion criteria are disputed, and there is no broadly accepted definition.

Gun Violence Archive, a nonprofit research group, run by Tracy Holtan, that tracks shootings and their characteristics in the United States, defines a mass shooting as an incident in which four or more people, excluding the perpetrator(s), are shot in one location at roughly the same time. The Congressional Research Service narrows that definition, limiting it to "public mass shootings", defined by four or more victims killed, excluding any victims who survive. The Washington Post and Mother Jones use similar definitions, with the latter acknowledging that their definition "is a conservative measure of the problem", as many shootings with fewer fatalities occur. The crowdsourced Mass Shooting Tracker project has the most expansive definition of four or more shot in any incident, including the perpetrator in the victim inclusion criteria.

A 2019 study of mass shootings published in the journal Injury Epidemiology recommended developing "a standard definition that considers both fatalities and nonfatalities to most appropriately convey the burden of mass shootings on gun violence." The authors of the study further suggested that "the definition of mass shooting should be four or more people, excluding the shooter, who are shot in a single event regardless of the motive, setting or number of deaths."

🔗 Computing 🔗 Computing/Computer hardware 🔗 Computing/Computer science

The replay system is a little-known subsystem within the Intel Pentium 4 processor. Its primary function is to catch operations that have been mistakenly sent for execution by the processor's scheduler. Operations caught by the replay system are then re-executed in a loop until the conditions necessary for their proper execution have been fulfilled.

🔗 Spaceflight 🔗 Environment 🔗 Disaster management

The Kessler syndrome (also called the Kessler effect, collisional cascading, or ablation cascade), proposed by NASA scientist Donald J. Kessler in 1978, is a theoretical scenario in which the density of objects in low Earth orbit (LEO) due to space pollution is high enough that collisions between objects could cause a cascade in which each collision generates space debris that increases the likelihood of further collisions. One implication is that the distribution of debris in orbit could render space activities and the use of satellites in specific orbital ranges difficult for many generations.

🔗 Astronomy 🔗 Astronomy/Astronomical objects

2024 YR₄ is an asteroid between 40 and 90 metres (130 and 300 ft) in diameter, classified as an Apollo-type (Earth-crossing) near-Earth object. It was discovered by the Chilean station of the Asteroid Terrestrial-impact Last Alert System (ATLAS) on 27 December 2024. As of 5 February 2025, 2024 YR₄ was rated 3 on the Torino scale with a 1 in 53 (1.9%) chance of impacting Earth on 22 December 2032. NASA gives a Palermo Technical Impact Hazard Scale rating of −0.40 for 2024 YR₄, which corresponds to an impact hazard of 39.8% of the background hazard level. The discovery has triggered the first step in planetary defense responses, in which all available telescopes are asked to gather data about the object and United Nations-endorsed space agencies are prompted to begin planning for asteroid threat mitigation.

Preliminary analysis of spectra and photometric timeseries of this asteroid suggests it is a stony S-type or L-type asteroid with a rotation period near 19.5 minutes. The asteroid previously made a close approach of 828,800 kilometres (515,000 miles; 2.156 lunar distances) to Earth on 25 December 2024 (two days before its discovery), and is now moving away from Earth. It will make its next close approach around 17 December 2028. By early April 2025 and until June 2028, 2024 YR₄ will have moved too far away from Earth to be observed by ground-based telescopes. Space-based infrared telescopes such as the James Webb Space Telescope may be able to observe 2024 YR₄ when it is far from Earth.

🔗 Computing

Lambda lifting is a meta-process that restructures a computer program so that functions are defined independently of each other in a global scope. An individual "lift" transforms a local function into a global function. It is a two step process, consisting of;

• Eliminating free variables in the function by adding parameters.
• Moving functions from a restricted scope to broader or global scope.

The term "lambda lifting" was first introduced by Thomas Johnsson around 1982 and was historically considered as a mechanism for implementing functional programming languages. It is used in conjunction with other techniques in some modern compilers.

Lambda lifting is not the same as closure conversion. It requires all call sites to be adjusted (adding extra arguments to calls) and does not introduce a closure for the lifted lambda expression. In contrast, closure conversion does not require call sites to be adjusted but does introduce a closure for the lambda expression mapping free variables to values.

The technique may be used on individual functions, in code refactoring, to make a function usable outside the scope in which it was written. Lambda lifts may also be repeated, in order to transform the program. Repeated lifts may be used to convert a program written in lambda calculus into a set of recursive functions, without lambdas. This demonstrates the equivalence of programs written in lambda calculus and programs written as functions. However it does not demonstrate the soundness of lambda calculus for deduction, as the eta reduction used in lambda lifting is the step that introduces cardinality problems into the lambda calculus, because it removes the value from the variable, without first checking that there is only one value that satisfies the conditions on the variable (see Curry's paradox).

Lambda lifting is expensive on processing time for the compiler. An efficient implementation of lambda lifting is ${\displaystyle O(n^{2})}$ on processing time for the compiler.

In the untyped lambda calculus, where the basic types are functions, lifting may change the result of beta reduction of a lambda expression. The resulting functions will have the same meaning, in a mathematical sense, but are not regarded as the same function in the untyped lambda calculus. See also intensional versus extensional equality.

The reverse operation to lambda lifting is lambda dropping.

Lambda dropping may make the compilation of programs quicker for the compiler, and may also increase the efficiency of the resulting program, by reducing the number of parameters, and reducing the size of stack frames. However it makes a function harder to re-use. A dropped function is tied to its context, and can only be used in a different context if it is first lifted.