Multicloud (also written as multi-cloud or multi cloud) is a term with varying interpretations, generally referring to a system using multiple cloud computing providers. According to ISO/IEC 22123-1: "multi-cloud is a cloud deployment model in which a customer uses public cloud services provided by two or more cloud service providers". Multi-cloud can involve various deployment models, including public, private, and hybrid clouds, and multiple service models, such as Infrastructure as a Service (IaaS), Platform as a Service (PaaS), and Software as a Service (SaaS). Multicloud incorporates workload, data, traffic, and workflow portability options, which can result in varying implementation complexity. When effectively implemented, multicloud solutions can enhance architectural resilience, reduce dependence on a single vendor, and improve flexibility by leveraging services from different providers. However, multicloud strategies also present challenges, including increased operational complexity, security risks, higher costs, and integration difficulties. According to the 2024 State of the Cloud Report by Flexera, multi-cloud adoption has continued to rise in 2024. Enterprises increasingly silo applications into specific clouds and select best-fit services. Key use cases include data analysis in separate clouds and cross-cloud disaster recovery. == Advantages and challenges == There are several advantages to using a multicloud approach, including the ability to negotiate better pricing with cloud providers, the ability to quickly switch to another provider if needed, and the ability to avoid vendor lock-in. Multicloud can also be a good way to hedge against the risks of obsolescence, as it allows you to rely on multiple vendors and open standards, which can prolong the life of your systems. Additional benefits of the multicloud architecture include adherence to local policies that require certain data to be physically present within the area/country, geographical distribution of processing requests from physically closer cloud unit which in turn reduces latency and protect against disasters. Various issues and challenges also present themselves in a multicloud environment. Security and governance is more complicated, and more "moving parts" may create resiliency issues. == Difference between multicloud and hybrid cloud == Multicloud differs from hybrid cloud in that it refers to multiple cloud services from different vendors rather than multiple deployment modes (on-premises hardware, and public and private, cloud hosting). However, when considering a broad definition of multi-cloud, hybrid cloud can still be regarded as a special form of multi-cloud.
Mistral Vibe
Mistral Vibe or Vibe (Le Chat until May 2026), is a chatbot that uses generative artificial intelligence developed in France by Mistral AI. Mistral Vibe is available in iOS and Android. Its services are operated on a freemium model. == History == In February 2024, Mistral AI released Le Chat. In January 2025, Mistral AI made a content deal with Agence France-Presse (AFP) that lets Le Chat query AFP's entire archive dating back to 1983. On 6 February 2025, a mobile app for Le Chat was released for iOS and Android, and a subscription tier, Pro, was introduced at a cost of $14.99 per month. In July 2025, Mistral AI released Voxtral, an open-source language model that understands and generates audio. Mistral introduced a voice mode for chatting that uses Voxtral, and projects, which allows grouping chats and files. In September 2025, Le Chat introduced the capability to remember previous conversations. In May 2026, Mistral AI announced the rebrand from Le Chat to Mistral Vibe and new features were introduced at the same time.
GPT-5
GPT-5 is a multimodal large language model developed by OpenAI and the fifth in its series of generative pre-trained transformer (GPT) foundation models. Preceded in the series by GPT-4, it was launched on August 7, 2025. It is publicly accessible to users of the chatbot products ChatGPT and Microsoft Copilot as well as to developers through the OpenAI API. == Background == On April 14, 2023, Sam Altman, the chief executive officer of OpenAI, spoke at an event at the Massachusetts Institute of Technology and said that the company was not training GPT-5 at that time. He stated that OpenAI was "prioritizing GPT-4 development" and that "we are not and won't for some time" release GPT-5. On July 18, OpenAI filed for a "GPT-5" trademark in the United States. On November 13, Altman confirmed to the Financial Times that the company was working to develop GPT-5. According to The Information, "[f]or much of the second half of 2024, OpenAI was developing a model known internally as Orion and intended to become GPT-5", "[b]ut the Orion effort failed to produce a better model, and the company instead released it as GPT-4.5 in February [2025]." By late July 2025, OpenAI was widely anticipated as planning to release GPT-5 in early August. On July 30, The Verge reported that "Microsoft is getting ready for GPT-5" as "sources familiar with Microsoft's AI plans" told an editor that the company was testing a new mode for its Copilot chatbot that would offer a model that "thinks deeply or quickly based on the task". On August 5, in the leadup to the release of GPT-5, OpenAI released GPT-OSS, a set of two open-weight models that have reasoning capabilities. GPT-5 was then unveiled during a livestream event on August 7. == Capabilities == At the time of its release, GPT-5 had state-of-the-art performance on benchmarks that test mathematics, programming, finance, and multimodal understanding. According to OpenAI, improvements over its predecessor models include faster response times, better coding and writing skills, more accurate answers to health questions, and lower levels of hallucination. Also, compared to previous models, GPT-5 aims to give safe, high-level responses to potentially harmful queries rather than outright declining them, an approach that OpenAI refers to as "safe completions", aiming to result "in GPT-5 being able to refuse more unsafe questions, while offering fewer rejections to users seeking harmless information." In addition, GPT-5 was trained to give more critical, "less effusively agreeable" answers compared to its predecessor models. Days before the launch of GPT-5, two early testers of the model stated that they were "impressed" by its ability to code and to solve mathematical and scientific problems. They suggested that the model shows great improvement from GPT-4, but not as large of a gain as from GPT-3 to GPT-4. A day prior to the release of GPT-5, during a press briefing, Sam Altman, the chief executive officer of OpenAI, called GPT-5 "a significant step along the path to AGI", referring to artificial general intelligence, the hypothetical level of intelligence that OpenAI defines as the ability to perform any economically valuable task that a human can. According to Altman, GPT-5 is "significantly better" than its predecessors, offering "PhD-level" abilities across a wide range of tasks. The exact energy consumption of GPT-5 use has not been disclosed by OpenAI. Researchers at the University of Rhode Island estimated that a medium-length response consumes slightly over 18 watt-hours, equivalent to using an incandescent bulb for 18 minutes. === Architecture === GPT-5 is a system that contains a fast, high-throughput model, a deeper reasoning model, and a real-time router that decides which model to use based on conversation type, complexity, tool needs, and explicit user intent. Altman had previously criticized the manual model picker for being overly complex, suggesting a need for unification. GPT-5 also includes agentic functionality through which it can set up its own desktop and can use its browser to search autonomously for sources that relate to its task. The GPT-5 system card defines two fast, high-throughput models – gpt-5-main and gpt-5-main-mini – and two thinking models – gpt-5-thinking and gpt-5-thinking-mini. In the OpenAI API, developers can access the thinking model, its mini version, and gpt-5-thinking-nano, an even smaller and faster nano version of the thinking model. The version of GPT-5 that is accessible via the API has adjustable reasoning effort (low, medium, high, or minimal) and verbosity (low, medium, or high). Additionally, ChatGPT provides access to gpt-5-thinking with a setting that makes use of parallel test-time compute, referred to as gpt-5-thinking-pro. == Limitations == === Safety === Neuraltrust, a security research company, claimed to have successfully compromised GPT-5 within its first day of testing the model. According to its report, it enabled GPT-5 to generate detailed instructions for manufacturing explosive devices. SPLX, another company, conducted similar tests and came to similar conclusions about GPT-5's security. Their assessments suggest that GPT-5 has significant security gaps, potentially rendering it as being unsafe for use in a corporate environment. == Training == According to AIMultiple, GPT-5 is natively multimodal, meaning that it was trained from scratch on multiple modalities (like text and images) at once without relying on already-trained language or vision models. Its training process involved three stages: unsupervised pretraining, supervised fine-tuning, and reinforcement learning from human feedback. Pretraining used a large-scale multilingual dataset of books, articles, web pages, academic papers, and licensed sources. GPT-5's visual and text capabilities were described as having been developed alongside each other throughout training, unlike with GPT-4. == Use == GPT-5 is used in ChatGPT. Although GPT-5 is free for all ChatGPT users, Plus users get higher use limits while Pro users get unlimited access to GPT-5 as well as limited access to GPT-5 Pro. Standard limits for lower-tier users on responses per hour still apply. Additionally, with the introduction of GPT-5, ChatGPT's "Advanced Voice Mode" was replaced by "ChatGPT Voice", which is supposed to enable more natural-sounding conversations. OpenAI stated that "Standard Voice Mode retires on September 9, 2025, unifying all users on ChatGPT Voice". On November 24, 2025, the feature of shopping research was added to ChatGPT, claimed to be a mini model post-trained on gpt-5-thinking-mini. GPT-5 is also available in Microsoft Copilot, and Microsoft stated that it will incorporate GPT-5 into a wide variety of its products. According to 9to5Mac, Apple Inc. is planning to integrate the model into the Apple Intelligence feature in its iOS 26, iPadOS 26, and macOS Tahoe operating systems. It is also accessible via the OpenAI API. A number of American companies were reported as having received access to GPT-5 ahead of its launch. OpenAI stated that the private health insurance company Oscar Health was checking applications from its policyholders with the model. In addition, Uber was using GPT-5 for its customer support system; GitLab, Windsurf, and Cursor were using the model for software development; and the Spanish bank BBVA was using it for financial analysis. Other companies that OpenAI listed as having used GPT-5 pre-release include Amgen, Lowe's, and Notion. == Reception == === Critical reviews === Grace Huckins in MIT Technology Review found that, "[w]hereas o1 was a major technological advancement, GPT-5 is, above all else, a refined product." In response to claims that Sam Altman, the chief executive officer of OpenAI, had made about the model, she stated that "GPT-5 will furnish a more pleasant and seamless user experience. That's not nothing, but it falls far short of the transformative AI future that Altman has spent much of the past year hyping." In response to Altman's claim that GPT-5 is "a significant step along the path" to artificial general intelligence, she noted: "[M]aybe he's right—but if so, it's a very small step." In The Information, Stephanie Palazzolo praised GPT-5's coding capabilities. According to Matteo Wong in The Atlantic, GPT-5 "is intuitive, fast, and efficient; adapts to human preferences and intentions; and is easy to personalize." He stated: "At this stage of the AI boom, when every major chatbot is legitimately helpful in numerous ways, benchmarks, science, and rigor feel almost insignificant. What matters is how the chatbot feels [...]". John Herrman from the New York magazine wrote: "Casual users who encounter GPT-5 through ChatGPT aren't likely to feel like they're using a completely different product [...] while people who use it for software development or in a corporate context are more likely to notice a major change." Mashable's Christian de Looper found that "GPT-5
PhyCV
PhyCV is the first computer vision library which utilizes algorithms directly derived from the equations of physics governing physical phenomena. The algorithms appearing in the first release emulate the propagation of light through a physical medium with natural and engineered diffractive properties followed by coherent detection. Unlike traditional algorithms that are a sequence of hand-crafted empirical rules, physics-inspired algorithms leverage physical laws of nature as blueprints. In addition, these algorithms can, in principle, be implemented in real physical devices for fast and efficient computation in the form of analog computing. Currently PhyCV has three algorithms, Phase-Stretch Transform (PST) and Phase-Stretch Adaptive Gradient-Field Extractor (PAGE), and Vision Enhancement via Virtual diffraction and coherent Detection (VEViD). All algorithms have CPU and GPU versions. PhyCV is now available on GitHub and can be installed from pip. == History == Algorithms in PhyCV are inspired by the physics of the photonic time stretch (a hardware technique for ultrafast and single-shot data acquisition). PST is an edge detection algorithm that was open-sourced in 2016 and has 800+ stars and 200+ forks on GitHub. PAGE is a directional edge detection algorithm that was open-sourced in February, 2022. PhyCV was originally developed and open-sourced by Jalali-Lab @ UCLA in May 2022. In the initial release of PhyCV, the original open-sourced code of PST and PAGE is significantly refactored and improved to be modular, more efficient, GPU-accelerated and object-oriented. VEViD is a low-light and color enhancement algorithm that was added to PhyCV in November 2022. == Background == === Phase-Stretch Transform (PST) === Phase-Stretch Transform (PST) is a computationally efficient edge and texture detection algorithm with exceptional performance in visually impaired images. The algorithm transforms the image by emulating propagation of light through a device with engineered diffractive property followed by coherent detection. It has been applied in improving the resolution of MRI image, extracting blood vessels in retina images, dolphin identification, and waste water treatment, single molecule biological imaging, and classification of UAV using micro Doppler imaging. === Phase-Stretch Adaptive Gradient-Field Extractor (PAGE) === Phase-Stretch Adaptive Gradient-Field Extractor (PAGE) is a physics-inspired algorithm for detecting edges and their orientations in digital images at various scales. The algorithm is based on the diffraction equations of optics. Metaphorically speaking, PAGE emulates the physics of birefringent (orientation-dependent) diffractive propagation through a physical device with a specific diffractive structure. The propagation converts a real-valued image into a complex function. Related information is contained in the real and imaginary components of the output. The output represents the phase of the complex function. === Vision Enhancement via Virtual diffraction and coherent Detection (VEViD) === Vision Enhancement via Virtual diffraction and coherent Detection (VEViD) an efficient and interpretable low-light and color enhancement algorithm that reimagines a digital image as a spatially varying metaphoric light field and then subjects the field to the physical processes akin to diffraction and coherent detection. The term “Virtual” captures the deviation from the physical world. The light field is pixelated and the propagation imparts a phase with an arbitrary dependence on frequency which can be different from the quadratic behavior of physical diffraction. VEViD can be further accelerated through mathematical approximations that reduce the computation time without appreciable sacrifice in image quality. A closed-form approximation for VEViD which we call VEViD-lite can achieve up to 200 FPS for 4K video enhancement. == PhyCV on the Edge == Featuring low-dimensionality and high-efficiency, PhyCV is ideal for edge computing applications. In this section, we demonstrate running PhyCV on NVIDIA Jetson Nano in real-time. === NVIDIA Jetson Nano Developer Kit === NVIDIA Jetson Nano Developer Kit is a small- sized and power-efficient platform for edge computing applications. It is equipped with an NVIDIA Maxwell architecture GPU with 128 CUDA cores, a quad-core ARM Cortex-A57 CPU, 4GB 64-bit LPDDR4 RAM, and supports video encoding and decoding up to 4K resolution. Jetson Nano also offers a variety of interfaces for connectivity and expansion, making it ideal for a wide range of AI and IoT applications. In our setup, we connect a USB camera to the Jetson Nano to acquire videos and demonstrate using PhyCV to process the videos in real-time. === Real-time PhyCV on Jetson Nano === We use the Jetson Nano (4GB) with NVIDIA JetPack SDK version 4.6.1, which comes with pre- installed Python 3.6, CUDA 10.2, and OpenCV 4.1.1. We further install PyTorch 1.10 to enable the GPU accelerated PhyCV. We demonstrate the results and metrics of running PhyCV on Jetson Nano in real-time for edge detection and low-light enhancement tasks. For 480p videos, both operations achieve beyond 38 FPS, which is sufficient for most cameras that capture videos at 30 FPS. For 720p videos, PhyCV low-light enhancement can operate at 24 FPS and PhyCV edge detection can operate at 17 FPS. == Highlights == === Modular Code Architecture === The code in PhyCV has a modular design which faithfully follows the physical process from which the algorithm was originated. Both PST and PAGE modules in the PhyCV library emulate the propagation of the input signal (original digital image) through a device with engineered diffractive property followed by coherent (phase) detection. The dispersive propagation applies a phase kernel to the frequency domain of the original image. This process has three steps in general, loading the image, initializing the kernel and applying the kernel. In the implementation of PhyCV, each algorithm is represented as a class in Python and each class has methods that simulate the steps described above. The modular code architecture follows the physics behind the algorithm. Please refer to the source code on GitHub for more details. === GPU Acceleration === PhyCV supports GPU acceleration. The GPU versions of PST and PAGE are built on PyTorch accelerated by the CUDA toolkit. The acceleration is beneficial for applying the algorithms in real-time image video processing and other deep learning tasks. The running time per frame of PhyCV algorithms on CPU (Intel i9-9900K) and GPU (NVIDIA TITAN RTX) for videos at different resolutions are shown below. Note that the PhyCV low-light enhancement operates in the HSV color space, so the running time also includes RGB to HSV conversion. However, for all running times using GPUs, we ignore the time of moving data from CPUs to GPUs and count the algorithm operation time only. == Installation and Examples == Please refer to the GitHub README file for a detailed technical documentation. == Current Limitations == === I/O (Input/Output) Bottleneck for Real-time Video Processing === When dealing with real-time video streams from cameras, the frames are captured and buffered in CPU and have to be moved to GPU to run the GPU-accelerated PhyCV algorithms. This process is time-consuming and it is a common bottleneck for real-time video-processing algorithms. === Lack of Parameter Adaptivity for Different Images === Currently, the parameters of PhyCV algorithms have to be manually tuned for different images. Although a set of pre-selected parameters work relatively well for a wide range of images, the lack of parameter adaptivity for different images remains a limitation for now.
Visual descriptor
In computer vision, visual descriptors or image descriptors are descriptions of the visual features of the contents in images, videos, or algorithms or applications that produce such descriptions. They describe elementary characteristics such as the shape, the color, the texture or the motion, among others. == Introduction == As a result of the new communication technologies and the massive use of Internet in our society, the amount of audio-visual information available in digital format is increasing considerably. Therefore, it has been necessary to design some systems that allow us to describe the content of several types of multimedia information in order to search and classify them. The audio-visual descriptors are in charge of the contents description. These descriptors have a good knowledge of the objects and events found in a video, image or audio and they allow the quick and efficient searches of the audio-visual content. This system can be compared to the search engines for textual contents. Although it is relatively easy to find text with a computer, it is much more difficult to find concrete audio and video parts. For instance, imagine somebody searching a scene of a happy person. The happiness is a feeling and it is not evident its shape, color and texture description in images. The description of the audio-visual content is not a superficial task and it is essential for the effective use of this type of archives. The standardization system that deals with audio-visual descriptors is the MPEG-7 (Motion Picture Expert Group - 7). == Types == Descriptors are the first step to find out the connection between pixels contained in a digital image and what humans recall after having observed an image or a group of images after some minutes. Visual descriptors are divided in two main groups: General information descriptors: contain low level descriptors which give a description about color, shape, regions, textures and motion. Specific domain information descriptors: give information about objects and events in the scene. A concrete example would be face recognition. === General information descriptors === General information descriptors consist of a set of descriptors that covers different basic and elementary features like: color, texture, shape, motion, location and others. This description is automatically generated by means of signal processing. ==== Color ==== It's the most basic quality of visual content. Five tools are defined to describe color. The three first tools represent the color distribution and the last ones describe the color relation between sequences or group of images: Dominant color descriptor (DCD) Scalable color descriptor (SCD) Color structure descriptor (CSD) Color layout descriptor (CLD) Group of frame (GoF) or group-of-pictures (GoP) ==== Texture ==== It's an important quality in order to describe an image. The texture descriptors characterize image textures or regions. They observe the region homogeneity and the histograms of these region borders. The set of descriptors is formed by: Homogeneous texture descriptor (HTD) Texture browsing descriptor (TBD) Edge histogram descriptor (EHD) ==== Shape ==== It contains important semantic information due to human's ability to recognize objects through their shape. However, this information can only be extracted by means of a segmentation similar to the one that the human visual system implements. Nowadays, such a segmentation system is not available yet, however there exists a serial of algorithms which are considered to be a good approximation. These descriptors describe regions, contours and shapes for 2D images and for 3D volumes. The shape descriptors are the following ones: Region-based shape descriptor (RSD) Contour-based shape descriptor (CSD) 3-D shape descriptor (3-D SD) ==== Motion ==== It's defined by four different descriptors which describe motion in video sequence. Motion is related to the objects motion in the sequence and to the camera motion. This last information is provided by the capture device, whereas the rest is implemented by means of image processing. The descriptor set is the following one: Motion activity descriptor (MAD) Camera motion descriptor (CMD) Motion trajectory descriptor (MTD) Warping and parametric motion descriptor (WMD and PMD) ==== Location ==== Elements location in the image is used to describe elements in the spatial domain. In addition, elements can also be located in the temporal domain: Region locator descriptor (RLD) Spatio temporal locator descriptor (STLD) === Specific domain information descriptors === These descriptors, which give information about objects and events in the scene, are not easily extractable, even more when the extraction is to be automatically done. Nevertheless, they can be manually processed. As mentioned before, face recognition is a concrete example of an application that tries to automatically obtain this information. == Descriptors applications == Among all applications, the most important ones are: Multimedia documents search engines and classifiers. Digital library: visual descriptors allow a very detailed and concrete search of any video or image by means of different search parameters. For instance, the search of films where a known actor appears, the search of videos containing the Everest mountain, etc. Personalized electronic news service. Possibility of an automatic connection to a TV channel broadcasting a soccer match, for example, whenever a player approaches the goal area. Control and filtering of concrete audiovisual content, like violent or pornographic material. Also, authorization for some multimedia content.
Symbolic regression
Symbolic regression (SR) is a type of regression analysis that searches the space of mathematical expressions to find the model that best fits a given dataset, both in terms of accuracy and simplicity. No particular model is provided as a starting point for symbolic regression. Instead, initial expressions are formed by randomly combining mathematical building blocks such as mathematical operators, analytic functions, constants, and state variables. Usually, a subset of these primitives will be specified by the person operating it, but that's not a requirement of the technique. The symbolic regression problem for mathematical functions has been tackled with a variety of methods, including recombining equations most commonly using genetic programming, as well as more recent methods utilizing Bayesian methods and neural networks. Another non-classical alternative method to SR is called Universal Functions Originator (UFO), which has a different mechanism, search-space, and building strategy. Further methods such as Exact Learning attempt to transform the fitting problem into a moments problem in a natural function space, usually built around generalizations of the Meijer-G function. By not requiring a priori specification of a model, symbolic regression isn't affected by human bias, or unknown gaps in domain knowledge. It attempts to uncover the intrinsic relationships of the dataset, by letting the patterns in the data itself reveal the appropriate models, rather than imposing a model structure that is deemed mathematically tractable from a human perspective. The fitness function that drives the evolution of the models takes into account not only error metrics (to ensure the models accurately predict the data), but also special complexity measures, thus ensuring that the resulting models reveal the data's underlying structure in a way that's understandable from a human perspective. This facilitates reasoning and favors the odds of getting insights about the data-generating system, as well as improving generalisability and extrapolation behaviour by preventing overfitting. Accuracy and simplicity may be left as two separate objectives of the regression—in which case the optimum solutions form a Pareto front—or they may be combined into a single objective by means of a model selection principle such as minimum description length. It has been proven that symbolic regression is an NP-hard problem. Nevertheless, if the sought-for equation is not too complex it is possible to solve the symbolic regression problem exactly by generating every possible function (built from some predefined set of operators) and evaluating them on the dataset in question. == Difference from classical regression == While conventional regression techniques seek to optimize the parameters for a pre-specified model structure, symbolic regression avoids imposing prior assumptions, and instead infers the model from the data. In other words, it attempts to discover both model structures and model parameters. This approach has the disadvantage of having a much larger space to search, because not only the search space in symbolic regression is infinite, but there are an infinite number of models which will perfectly fit a finite data set (provided that the model complexity isn't artificially limited). This means that it will possibly take a symbolic regression algorithm longer to find an appropriate model and parametrization, than traditional regression techniques. This can be attenuated by limiting the set of building blocks provided to the algorithm, based on existing knowledge of the system that produced the data; but in the end, using symbolic regression is a decision that has to be balanced with how much is known about the underlying system. Nevertheless, this characteristic of symbolic regression also has advantages: because the evolutionary algorithm requires diversity in order to effectively explore the search space, the result is likely to be a selection of high-scoring models (and their corresponding set of parameters). Examining this collection could provide better insight into the underlying process, and allows the user to identify an approximation that better fits their needs in terms of accuracy and simplicity. == Benchmarking == === SRBench === In 2021, SRBench was proposed as a large benchmark for symbolic regression. In its inception, SRBench featured 14 symbolic regression methods, 7 other ML methods, and 252 datasets from PMLB. The benchmark intends to be a living project: it encourages the submission of improvements, new datasets, and new methods, to keep track of the state of the art in SR. === SRBench Competition 2022 === In 2022, SRBench announced the competition Interpretable Symbolic Regression for Data Science, which was held at the GECCO conference in Boston, MA. The competition pitted nine leading symbolic regression algorithms against each other on a novel set of data problems and considered different evaluation criteria. The competition was organized in two tracks, a synthetic track and a real-world data track. ==== Synthetic Track ==== In the synthetic track, methods were compared according to five properties: re-discovery of exact expressions; feature selection; resistance to local optima; extrapolation; and sensitivity to noise. Rankings of the methods were: QLattice PySR (Python Symbolic Regression) uDSR (Deep Symbolic Optimization) ==== Real-world Track ==== In the real-world track, methods were trained to build interpretable predictive models for 14-day forecast counts of COVID-19 cases, hospitalizations, and deaths in New York State. These models were reviewed by a subject expert and assigned trust ratings and evaluated for accuracy and simplicity. The ranking of the methods was: uDSR (Deep Symbolic Optimization) QLattice geneticengine (Genetic Engine) == Non-standard methods == Most symbolic regression algorithms prevent combinatorial explosion by implementing evolutionary algorithms that iteratively improve the best-fit expression over many generations. Recently, researchers have proposed algorithms utilizing other tactics in AI. Silviu-Marian Udrescu and Max Tegmark developed the "AI Feynman" algorithm, which attempts symbolic regression by training a neural network to represent the mystery function, then runs tests against the neural network to attempt to break up the problem into smaller parts. For example, if f ( x 1 , . . . , x i , x i + 1 , . . . , x n ) = g ( x 1 , . . . , x i ) + h ( x i + 1 , . . . , x n ) {\displaystyle f(x_{1},...,x_{i},x_{i+1},...,x_{n})=g(x_{1},...,x_{i})+h(x_{i+1},...,x_{n})} , tests against the neural network can recognize the separation and proceed to solve for g {\displaystyle g} and h {\displaystyle h} separately and with different variables as inputs. This is an example of divide and conquer, which reduces the size of the problem to be more manageable. AI Feynman also transforms the inputs and outputs of the mystery function in order to produce a new function which can be solved with other techniques, and performs dimensional analysis to reduce the number of independent variables involved. The algorithm was able to "discover" 100 equations from The Feynman Lectures on Physics, while a leading software using evolutionary algorithms, Eureqa, solved only 71. AI Feynman, in contrast to classic symbolic regression methods, requires a very large dataset in order to first train the neural network and is naturally biased towards equations that are common in elementary physics.
Graph cut optimization
Graph cut optimization is a combinatorial optimization method applicable to a family of functions of discrete variables, named after the concept of cut in the theory of flow networks. Thanks to the max-flow min-cut theorem, determining the minimum cut over a graph representing a flow network is equivalent to computing the maximum flow over the network. Given a pseudo-Boolean function f {\displaystyle f} , if it is possible to construct a flow network with positive weights such that each cut C {\displaystyle C} of the network can be mapped to an assignment of variables x {\displaystyle \mathbf {x} } to f {\displaystyle f} (and vice versa), and the cost of C {\displaystyle C} equals f ( x ) {\displaystyle f(\mathbf {x} )} (up to an additive constant) then it is possible to find the global optimum of f {\displaystyle f} in polynomial time by computing a minimum cut of the graph. The mapping between cuts and variable assignments is done by representing each variable with one node in the graph and, given a cut, each variable will have a value of 0 if the corresponding node belongs to the component connected to the source, or 1 if it belong to the component connected to the sink. Not all pseudo-Boolean functions can be represented by a flow network, and in the general case the global optimization problem is NP-hard. There exist sufficient conditions to characterise families of functions that can be optimised through graph cuts, such as submodular quadratic functions. Graph cut optimization can be extended to functions of discrete variables with a finite number of values, that can be approached with iterative algorithms with strong optimality properties, computing one graph cut at each iteration. Graph cut optimization is an important tool for inference over graphical models such as Markov random fields or conditional random fields, and it has applications in computer vision problems such as image segmentation, denoising, registration and stereo matching. == Representability == A pseudo-Boolean function f : { 0 , 1 } n → R {\displaystyle f:\{0,1\}^{n}\to \mathbb {R} } is said to be representable if there exists a graph G = ( V , E ) {\displaystyle G=(V,E)} with non-negative weights and with source and sink nodes s {\displaystyle s} and t {\displaystyle t} respectively, and there exists a set of nodes V 0 = { v 1 , … , v n } ⊂ V − { s , t } {\displaystyle V_{0}=\{v_{1},\dots ,v_{n}\}\subset V-\{s,t\}} such that, for each tuple of values ( x 1 , … , x n ) ∈ { 0 , 1 } n {\displaystyle (x_{1},\dots ,x_{n})\in \{0,1\}^{n}} assigned to the variables, f ( x 1 , … , x n ) {\displaystyle f(x_{1},\dots ,x_{n})} equals (up to a constant) the value of the flow determined by a minimum cut C = ( S , T ) {\displaystyle C=(S,T)} of the graph G {\displaystyle G} such that v i ∈ S {\displaystyle v_{i}\in S} if x i = 0 {\displaystyle x_{i}=0} and v i ∈ T {\displaystyle v_{i}\in T} if x i = 1 {\displaystyle x_{i}=1} . It is possible to classify pseudo-Boolean functions according to their order, determined by the maximum number of variables contributing to each single term. All first order functions, where each term depends upon at most one variable, are always representable. Quadratic functions f ( x ) = w 0 + ∑ i w i ( x i ) + ∑ i < j w i j ( x i , x j ) . {\displaystyle f(\mathbf {x} )=w_{0}+\sum _{i}w_{i}(x_{i})+\sum _{i