Mean shift is a non-parametric feature-space mathematical analysis technique for locating the maxima of a density function, a so-called mode-seeking algorithm. Application domains include cluster analysis in computer vision and image processing. == History == The mean shift procedure is usually credited to work by Fukunaga and Hostetler in 1975. It is, however, reminiscent of earlier work by Schnell in 1964. == Overview == Mean shift is a procedure for locating the maxima—the modes—of a density function given discrete data sampled from that function. This is an iterative method, and we start with an initial estimate x {\displaystyle x} . Let a kernel function K ( x i − x ) {\displaystyle K(x_{i}-x)} be given. This function determines the weight of nearby points for re-estimation of the mean. Typically a Gaussian kernel on the distance to the current estimate is used, K ( x i − x ) = e − c | | x i − x | | 2 {\displaystyle K(x_{i}-x)=e^{-c||x_{i}-x||^{2}}} . The weighted mean of the density in the window determined by K {\displaystyle K} is m ( x ) = ∑ x i ∈ N ( x ) K ( x i − x ) x i ∑ x i ∈ N ( x ) K ( x i − x ) {\displaystyle m(x)={\frac {\sum _{x_{i}\in N(x)}K(x_{i}-x)x_{i}}{\sum _{x_{i}\in N(x)}K(x_{i}-x)}}} where N ( x ) {\displaystyle N(x)} is the neighborhood of x {\displaystyle x} , a set of points for which K ( x i − x ) ≠ 0 {\displaystyle K(x_{i}-x)\neq 0} . The difference m ( x ) − x {\displaystyle m(x)-x} is called mean shift in Fukunaga and Hostetler. The mean-shift algorithm now sets x ← m ( x ) {\displaystyle x\leftarrow m(x)} , and repeats the estimation until m ( x ) {\displaystyle m(x)} converges. Although the mean shift algorithm has been widely used in many applications, a rigid proof for the convergence of the algorithm using a general kernel in a high dimensional space is still not known. Aliyari Ghassabeh showed the convergence of the mean shift algorithm in one dimension with a differentiable, convex, and strictly decreasing profile function. However, the one-dimensional case has limited real world applications. Also, the convergence of the algorithm in higher dimensions with a finite number of the stationary (or isolated) points has been proved. However, sufficient conditions for a general kernel function to have finite stationary (or isolated) points have not been provided. Gaussian Mean-Shift is an Expectation–maximization algorithm. == Details == Let data be a finite set S {\displaystyle S} embedded in the n {\displaystyle n} -dimensional Euclidean space, X {\displaystyle X} . Let K {\displaystyle K} be a flat kernel that is the characteristic function of the λ {\displaystyle \lambda } -ball in X {\displaystyle X} , In each iteration of the algorithm, s ← m ( s ) {\displaystyle s\leftarrow m(s)} is performed for all s ∈ S {\displaystyle s\in S} simultaneously. The first question, then, is how to estimate the density function given a sparse set of samples. One of the simplest approaches is to just smooth the data, e.g., by convolving it with a fixed kernel of width h {\displaystyle h} , where x i {\displaystyle x_{i}} are the input samples and k ( r ) {\displaystyle k(r)} is the kernel function (or Parzen window). h {\displaystyle h} is the only parameter in the algorithm and is called the bandwidth. This approach is known as kernel density estimation or the Parzen window technique. Once we have computed f ( x ) {\displaystyle f(x)} from the equation above, we can find its local maxima using gradient ascent or some other optimization technique. The problem with this "brute force" approach is that, for higher dimensions, it becomes computationally prohibitive to evaluate f ( x ) {\displaystyle f(x)} over the complete search space. Instead, mean shift uses a variant of what is known in the optimization literature as multiple restart gradient descent. Starting at some guess for a local maximum, y k {\displaystyle y_{k}} , which can be a random input data point x 1 {\displaystyle x_{1}} , mean shift computes the gradient of the density estimate f ( x ) {\displaystyle f(x)} at y k {\displaystyle y_{k}} and takes an uphill step in that direction. == Types of kernels == Kernel definition: Let X {\displaystyle X} be the n {\displaystyle n} -dimensional Euclidean space, R n {\displaystyle \mathbb {R} ^{n}} . The norm of x {\displaystyle x} is a non-negative number, ‖ x ‖ 2 = x ⊤ x ≥ 0 {\displaystyle \|x\|^{2}=x^{\top }x\geq 0} . A function K : X → R {\displaystyle K:X\rightarrow \mathbb {R} } is said to be a kernel if there exists a profile, k : [ 0 , ∞ ] → R {\displaystyle k:[0,\infty ]\rightarrow \mathbb {R} } , such that K ( x ) = k ( ‖ x ‖ 2 ) {\displaystyle K(x)=k(\|x\|^{2})} and k is non-negative. k is non-increasing: k ( a ) ≥ k ( b ) {\displaystyle k(a)\geq k(b)} if a < b {\displaystyle a ZygoteBody, formerly Google Body, is a web application by Zygote Media Group that renders manipulable 3D anatomical models of the human body. Several layers, from muscle tissues down to blood vessels, can be removed or made transparent to allow better study of individual body parts. Most of the body parts are labelled and are searchable. == Technology == The human models are based on data from the Zygote Media Group. The website uses JavaScript and WebGL technology to display 3D images inside the web browser without requiring the installation of external browser plug-ins. == History == ZygoteBody was launched as Google Body on December 15, 2010. On April Fools' Day 2011, users were greeted with the anatomy of a cow on the home page. The cow model is still available as part of the open-3d-viewer open source project. As part of the wind down on Google Labs, it was announced that Google Body will be shut down but will continue to be maintained by Zygote as ZygoteBody. On October 13, 2011, the Google Body site was shut down. Then, on January 9, 2012, ZygoteBody was launched and core code base (with the Google Cow model as a demo) was made available as an open source project called open-3d-viewer. In cryptography, the hybrid argument is a proof technique used to show that two distributions are computationally indistinguishable. == History == Hybrid arguments had their origin in a papers by Andrew Yao in 1982 and Shafi Goldwasser and Silvio Micali in 1983. == Formal description == Formally, to show two distributions D1 and D2 are computationally indistinguishable, we can define a sequence of hybrid distributions D1 := H0, H1, ..., Ht =: D2 where t is polynomial in the security parameter n. Define the advantage of any probabilistic efficient (polynomial-bounded time) algorithm A as A d v H i , H i + 1 d i s t ( A ) := | Pr [ x ← $ H i : A ( x ) = 1 ] − Pr [ x ← $ H i + 1 : A ( x ) = 1 ] | , {\displaystyle {\mathsf {Adv}}_{H_{i},H_{i+1}}^{\mathsf {dist}}(\mathbf {A} ):=\left|\Pr[x{\stackrel {\$}{\gets }}H_{i}:\mathbf {A} (x)=1]-\Pr[x{\stackrel {\$}{\gets }}H_{i+1}:\mathbf {A} (x)=1]\right|,} where the dollar symbol ($) denotes that we sample an element from the distribution at random. By triangle inequality, it is clear that for any probabilistic polynomial time algorithm A, A d v D 1 , D 2 d i s t ( A ) ≤ ∑ i = 0 t − 1 A d v H i , H i + 1 d i s t ( A ) . {\displaystyle {\mathsf {Adv}}_{D_{1},D_{2}}^{\mathsf {dist}}(\mathbf {A} )\leq \sum _{i=0}^{t-1}{\mathsf {Adv}}_{H_{i},H_{i+1}}^{\mathsf {dist}}(\mathbf {A} ).} Thus there must exist some k s.t. 0 ≤ k < t(n) and A d v H k , H k + 1 d i s t ( A ) ≥ A d v D 1 , D 2 d i s t ( A ) / t ( n ) . {\displaystyle {\mathsf {Adv}}_{H_{k},H_{k+1}}^{\mathsf {dist}}(\mathbf {A} )\geq {\mathsf {Adv}}_{D_{1},D_{2}}^{\mathsf {dist}}(\mathbf {A} )/t(n).} Since t is polynomial-bounded, for any such algorithm A, if we can show that it has a fixed negligible advantage function ε(n) between distributions Hi and Hi+1 for every i, so in particular, ϵ ( n ) ≥ A d v H k , H k + 1 d i s t ( A ) ≥ A d v D 1 , D 2 d i s t ( A ) / t ( n ) , {\displaystyle \epsilon (n)\geq {\mathsf {Adv}}_{H_{k},H_{k+1}}^{\mathsf {dist}}(\mathbf {A} )\geq {\mathsf {Adv}}_{D_{1},D_{2}}^{\mathsf {dist}}(\mathbf {A} )/t(n),} then it immediately follows that its advantage to distinguish the distributions D1 = H0 and D2 = Ht must also be negligible. == Applications == The hybrid argument is extensively used in cryptography. Some simple proofs using hybrid arguments are: If one cannot efficiently predict the next bit of the output of some number generator, then this generator is a pseudorandom number generator (PRG). We can securely expand a PRG with 1-bit output into a PRG with n-bit output. Key Transparency allows communicating parties to verify public keys used in end-to-end encryption. In many end-to-end encryption services, to initiate communication a user will reach out to a central server and request the public keys of the user with which they wish to communicate. If the central server is malicious or becomes compromised, a man-in-the-middle attack can be launched through the issuance of incorrect public keys. The communications can then be intercepted and manipulated. Additionally, legal pressure could be applied by surveillance agencies to manipulate public keys and read messages. With Key Transparency, public keys are posted to a public log that can be universally audited. Communicating parties can verify public keys used are accurate. Data proliferation refers to the prodigious amount of data, structured and unstructured, that businesses and governments continue to generate at an unprecedented rate and the usability problems that result from attempting to store and manage that data. While originally pertaining to problems associated with paper documentation, data proliferation has become a major problem in primary and secondary data storage on computers. While digital storage has become cheaper, the associated costs, from raw power to maintenance and from metadata to search engines, have not kept up with the proliferation of data. Although the power required to maintain a unit of data has fallen, the cost of facilities which house the digital storage has tended to rise. Data proliferation has been documented as a problem for the U.S. military since August 1971, in particular regarding the excessive documentation submitted during the acquisition of major weapon systems. Efforts to mitigate data proliferation and the problems associated with it are ongoing. == Problems caused == The problem of data proliferation is affecting all areas of commerce as a result of the availability of relatively inexpensive data storage devices. This has made it very easy to dump data into secondary storage immediately after its window of usability has passed. This masks problem that could gravely affect the profitability of businesses and the efficient functioning of health services, police and security forces, local and national governments, and many other types of organizations. Data proliferation is problematic for several reasons: Difficulty when trying to find and retrieve information. At Xerox, on average it takes employees more than one hour per week to find hard-copy documents, costing $2,152 a year to manage and store them. For businesses with more than 10 employees, this increases to almost two hours per week at $5,760 per year. In large networks of primary and secondary data storage, problems finding electronic data are analogous to problems finding hard copy data. Data loss and legal liability when data is disorganized, not properly replicated, or cannot be found promptly. In April 2005, the Ameritrade Holding Corporation told 200,000 current and past customers that a tape containing confidential information had been lost or destroyed in transit. In May of the same year, Time Warner Incorporated reported that 40 tapes containing personal data on 600,000 current and former employees had been lost en route to a storage facility. In March 2005, a Florida judge hearing a $2.7 billion lawsuit against Morgan Stanley issued an "adverse inference order" against the company for "willful and gross abuse of its discovery obligations." The judge cited Morgan Stanley for repeatedly finding misplaced tapes of e-mail messages long after the company had claimed that it had turned over all such tapes to the court. Increased manpower requirements to manage increasingly chaotic data storage resources. Slower networks and application performance due to excess traffic as users search and search again for the material they need. High cost in terms of the energy resources required to operate storage hardware. A 100 terabyte system will cost up to $35,040 a year to run—not counting cooling costs. == Proposed solutions == Applications that better utilize modern technology Reductions in duplicate data (especially as caused by data movement) Improvement of metadata structures Improvement of file and storage transfer structures User education and discipline The implementation of Information Lifecycle Management solutions to eliminate low-value information as early as possible before putting the rest into actively managed long-term storage in which it can be quickly and cheaply accessed. XLA (Accelerated Linear Algebra) is an open-source compiler for machine learning developed by the OpenXLA project. XLA is designed to improve the performance of machine learning models by optimizing the computation graphs at a lower level, making it particularly useful for large-scale computations and high-performance machine learning models. Key features of XLA include: Compilation of Computation Graphs: Compiles computation graphs into efficient machine code. Optimization Techniques: Applies operation fusion, memory optimization, and other techniques. Hardware Support: Optimizes models for various hardware, including CPUs, GPUs, and NPUs. Improved Model Execution Time: Aims to reduce machine learning models' execution time for both training and inference. Seamless Integration: Can be used with existing machine learning code with minimal changes. XLA represents a significant step in optimizing machine learning models, providing developers with tools to enhance computational efficiency and performance. == OpenXLA Project == OpenXLA Project is an open-source machine learning compiler and infrastructure initiative intended to provide a common set of tools for compiling and deploying machine learning models across different frameworks and hardware platforms. It provides a modular compilation stack that can be used by major deep learning frameworks like JAX, PyTorch, and TensorFlow. The project focuses on supplying shared components for optimization, portability, and execution across CPUs, GPUs, and specialized accelerators. Its design emphasizes interoperability between frameworks and a standardized set of representations for model computation. == Components == The OpenXLA ecosystem includes several core components: XLA – A deep learning compiler that optimizes computational graphs for multiple hardware targets. PJRT – A runtime interface that allows different back-ends to connect to XLA through a consistent API. StableHLO – A high-level operator set intended to serve as a stable, portable representation for ML models across compilers and frameworks. Shardy – An MLIR-based system for describing and transforming models that run in distributed or multi-device environments. Additional profiling, testing, and integration tools maintained under the OpenXLA organization. == Users and adopters == Several machine learning frameworks can use or interoperate with OpenXLA components, including JAX, TensorFlow, and parts of the PyTorch ecosystem. The project is developed with participation from multiple hardware and software organizations that contribute back-end integrations, testing, or specifications for their devices. This includes Alibaba, Amazon Web Services, AMD, Anyscale, Apple, Arm, Cerebras, Google, Graphcore, Hugging Face, Intel, Meta, NVIDIA and SiFive. == Supported target devices == x86-64 ARM64 NVIDIA GPU AMD GPU Intel GPU Apple GPU Google TPU AWS Trainium, Inferentia Cerebras Graphcore IPU == Governance == OpenXLA is developed as a community project with its work carried out in public repositories, discussion forums, and design meetings. Some components, such as StableHLO, began with stewardship from specific organizations and have outlined plans for more formal and distributed governance models as the project matures. == History == The project was announced in 2022 as an effort to coordinate development of ML compiler technologies across major AI companies, notably: Alibaba, Amazon Web Services, AMD, Anyscale, Apple, Arm, Cerebras, Google, Graphcore, Hugging Face, Intel, Meta, NVIDIA and SiFive.. It consolidated the XLA compiler, introduced StableHLO as a portable operator set, and created a unified structure for additional tools. Development continues within multiple repositories under the OpenXLA umbrella. It was founded by Eugene Burmako, James Rubin, Magnus Hyttsten, Mehdi Amini, Navid Khajouei, and Thea Lamkin from Google's Machine Learning organization. A data dictionary, or metadata repository, as defined in the IBM Dictionary of Computing, is a "centralized repository of information about data such as meaning, relationships to other data, origin, usage, and format". Oracle defines it as a collection of tables with metadata. The term can have one of several closely related meanings pertaining to databases and database management systems (DBMS): A document describing a database or collection of databases An integral component of a DBMS that is required to determine its structure A piece of middleware that extends or supplants the native data dictionary of a DBMS == Documentation == The terms data dictionary and data repository indicate a more general software utility than a catalogue. A catalogue is closely coupled with the DBMS software. It provides the information stored in it to the user and the DBA, but it is mainly accessed by the various software modules of the DBMS itself, such as DDL and DML compilers, the query optimiser, the transaction processor, report generators, and the constraint enforcer. On the other hand, a data dictionary is a data structure that stores metadata, i.e., (structured) data about information. The software package for a stand-alone data dictionary or data repository may interact with the software modules of the DBMS, but it is mainly used by the designers, users and administrators of a computer system for information resource management. These systems maintain information on system hardware and software configuration, documentation, application and users as well as other information relevant to system administration. If a data dictionary system is used only by the designers, users, and administrators and not by the DBMS Software, it is called a passive data dictionary. Otherwise, it is called an active data dictionary or data dictionary. When a passive data dictionary is updated, it is done so manually and independently from any changes to a DBMS (database) structure. With an active data dictionary, the dictionary is updated first and changes occur in the DBMS automatically as a result. Database users and application developers can benefit from an authoritative data dictionary document that catalogs the organization, contents, and conventions of one or more databases. This typically includes the names and descriptions of various tables (records or entities) and their contents (fields), plus additional details, like the type and length of each data element. Another important piece of information that a data dictionary can provide is the relationship between tables. This is sometimes referred to in entity-relationship diagrams (ERDs), or if using set descriptors, identifying which sets database tables participate in. In an active data dictionary constraints may be placed upon the underlying data. For instance, a range may be imposed on the value of numeric data in a data element (field), or a record in a table may be forced to participate in a set relationship with another record-type. Additionally, a distributed DBMS may have certain location specifics described within its active data dictionary (e.g. where tables are physically located). The data dictionary consists of record types (tables) created in the database by systems generated command files, tailored for each supported back-end DBMS. Oracle has a list of specific views for the "sys" user. This allows users to look up the exact information that is needed. Command files contain SQL Statements for CREATE TABLE, CREATE UNIQUE INDEX, ALTER TABLE (for referential integrity), etc., using the specific statement required by that type of database. There is no universal standard as to the level of detail in such a document. == Middleware == In the construction of database applications, it can be useful to introduce an additional layer of data dictionary software, i.e. middleware, which communicates with the underlying DBMS data dictionary. Such a "high-level" data dictionary may offer additional features and a degree of flexibility that goes beyond the limitations of the native "low-level" data dictionary, whose primary purpose is to support the basic functions of the DBMS, not the requirements of a typical application. For example, a high-level data dictionary can provide alternative entity-relationship models tailored to suit different applications that share a common database. Extensions to the data dictionary also can assist in query optimization against distributed databases. Additionally, DBA functions are often automated using restructuring tools that are tightly coupled to an active data dictionary. Software frameworks aimed at rapid application development sometimes include high-level data dictionary facilities, which can substantially reduce the amount of programming required to build menus, forms, reports, and other components of a database application, including the database itself. For example, PHPLens includes a PHP class library to automate the creation of tables, indexes, and foreign key constraints portably for multiple databases. Another PHP-based data dictionary, part of the RADICORE toolkit, automatically generates program objects, scripts, and SQL code for menus and forms with data validation and complex joins. For the ASP.NET environment, Base One's data dictionary provides cross-DBMS facilities for automated database creation, data validation, performance enhancement (caching and index utilization), application security, and extended data types. Visual DataFlex features provides the ability to use DataDictionaries as class files to form middle layer between the user interface and the underlying database. The intent is to create standardized rules to maintain data integrity and enforce business rules throughout one or more related applications. Some industries use generalized data dictionaries as technical standards to ensure interoperability between systems. The real estate industry, for example, abides by a RESO's Data Dictionary to which the National Association of REALTORS mandates its MLSs comply with through its policy handbook. This intermediate mapping layer for MLSs' native databases is supported by software companies which provide API services to MLS organizations. == Platform-specific examples == Developers use a data description specification (DDS) to describe data attributes in file descriptions that are external to the application program that processes the data, in the context of an IBM i. The sys.ts$ table in Oracle stores information about every table in the database. It is part of the data dictionary that is created when the Oracle Database is created. Developers may also use DDS context from free and open-source software (FOSS) for structured and transactional queries in open environments. == Typical attributes == Here is a non-exhaustive list of typical items found in a data dictionary for columns or fields: Entity or form name or their ID (EntityID or FormID). The group this field belongs to. Field name, such as RDBMS field name Displayed field title. May default to field name if blank. Field type (string, integer, date, etc.) Measures such as min and max values, display width, or number of decimal places. Different field types may interpret this differently. An alternative is to have different attributes depending on field type. Field display order or tab order Coordinates on screen (if a positional or grid-based UI) Default value Prompt type, such as drop-down list, combo-box, check-boxes, range, etc. Is-required (Boolean) - If 'true', the value cannot be blank, null, or only white-spaces Is-read-only (Boolean) Reference table name, if a foreign key. Can be used for validation or selection lists. Various event handlers or references to. Example: "on-click", "on-validate", etc. See event-driven programming. Format code, such as a regular expression or COBOL-style "PIC" statements Description or synopsis Database index characteristics or specificationZygoteBody
Hybrid argument (cryptography)
Key Transparency
Data proliferation
Accelerated Linear Algebra
Data dictionary