In mathematical psychology and education theory, a knowledge space is a combinatorial structure used to formulate mathematical models describing the progression of a human learner. Knowledge spaces were introduced in 1985 by Jean-Paul Doignon and Jean-Claude Falmagne, and remain in extensive use in the education theory. Modern applications include two computerized tutoring systems, ALEKS and the defunct RATH. Formally, a knowledge space assumes that a domain of knowledge is a collection of concepts or skills, each of which must be eventually mastered. Not all concepts are interchangeable; some require other concepts as prerequisites. Conversely, competency at one skill may ease the acquisition of another through similarity. A knowledge space marks out which collections of skills are feasible: they can be learned without mastering any other skills. Under reasonable assumptions, the collection of feasible competencies forms the mathematical structure known as an antimatroid. Researchers and educators usually explore the structure of a discipline's knowledge space as a latent class model. == Motivation == Knowledge Space Theory attempts to address shortcomings of standardized testing when used in educational psychometry. Common tests, such as the SAT and ACT, compress a student's knowledge into a very small range of ordinal ranks, in the process effacing the conceptual dependencies between questions. Consequently, the tests cannot distinguish between true understanding and guesses, nor can they identify a student's particular weaknesses, only the general proportion of skills mastered. The goal of knowledge space theory is to provide a language by which exams can communicate What the student can do and What the student is ready to learn. == Model structure == Knowledge Space Theory-based models presume that an educational subject S can be modeled as a finite set Q of concepts, skills, or topics. Each feasible state of knowledge about S is then a subset of Q; the set of all such feasible states is K. The precise term for the information (Q, K) depends on the extent to which K satisfies certain axioms: A knowledge structure assumes that K contains the empty set (a student may know nothing about S) and Q itself (a student may have fully mastered S). A knowledge space is a knowledge structure that is closed under set union: if, for each topic, there is an expert in a class on that topic, then it is possible, with enough time and effort, for each student in the class to become an expert on all those topics simultaneously. A quasi-ordinal knowledge space is a knowledge space that is also closed under set intersection: if student a knows topics A and B; and student c knows topics B and C; then it is possible for another student b to know only topic B. A well-graded knowledge space or learning space is a knowledge space satisfying the following axiom: If S∈K, then there exists x∈S such that S\{x}∈K In educational terms, any feasible body of knowledge can be learned one concept at a time. === Prerequisite partial order === The more contentful axioms associated with quasi-ordinal and well-graded knowledge spaces each imply that the knowledge space forms a well-understood (and heavily studied) mathematical structure: A quasi-ordinal knowledge space can be associated with a distributive lattice under set union and set intersection. The name "quasi-ordinal" arises from Birkhoff's representation theorem, which explains that distributive lattices uniquely correspond to partial orders. A well-graded knowledge space is an antimatroid, a type of mathematical structure that describes certain problems solvable with a greedy algorithm. In either case, the mathematical structure implies that set inclusion defines partial order on K, interpretable as an educational prerequirement: if a(⪯)b in this partial order, then a must be learned before b. === Inner and outer fringe === The prerequisite partial order does not uniquely identify a curriculum; some concepts may lead to a variety of other possible topics. But the covering relation associated with the prerequisite partial does control curricular structure: if students know a before a lesson and b immediately after, then b must cover a in the partial order. In such a circumstance, the new topics covered between a and b constitute the outer fringe of a ("what the student was ready to learn") and the inner fringe of b ("what the student just learned"). == Construction of knowledge spaces == In practice, there exist several methods to construct knowledge spaces. The most frequently used method is querying experts. There exist several querying algorithms that allow one or several experts to construct a knowledge space by answering a sequence of simple questions. Another method is to construct the knowledge space by explorative data analysis (for example by item tree analysis) from data. A third method is to derive the knowledge space from an analysis of the problem solving processes in the corresponding domain.
Video Super Resolution
RTX Video Super Resolution (RTX VSR) is a video scaling feature by Nvidia. It was released on February 28, 2023. == History == The feature was first unveiled during CES 2023 as RTX Video Super Resolution. It uses the on-board Tensor Cores to upscale browser video content in real time. Video Super Resolution was initially only available on RTX 30 and 40 series GPUs, while support for 20 series GPUs was added afterwards; it is now available on all Nvidia RTX-branded GPUs. The feature supports input resolutions from 360p to 1440p and a max output of 4K and comes without support for HDR content although that could be likely added in the future. Nvidia released RTX Video Super Resolution 1.5 with improved video quality and RTX 20 series support on October 17, 2023. == Reception == According to ComputerBase, although "the algorithm is not yet working flawlessly", the feature is "overall recommendable".
List of 3D rendering software
3D rendering software products are the dedicated engines used for rendering computer-generated imagery. This is not the same as 3D modeling software, which involves the creation of 3D models, for which the software listed below can produce realistically rendered visualisations.General-purpose packages which can have their own built-in rendering capabilities are not listed here; these can be found in the list of 3D computer graphics software and list of 3D animation software. See 3D computer graphics software for more discussion about the distinctions.
Breakup Notifier
Breakup Notifier was a web application written by product developer and programmer Dan Loewenherz that enabled its registered users to track the relationship status of their Facebook friends. An email notification was sent to the user when one of their Facebook friends changed their relationship status. The app was one of the most viral Facebook app's at the time of its release. It was mentioned in a skit on The Jay Leno Show and news of its popularity was published in Time magazine, The New York Post, CNET, and The Globe and Mail. == Popularity and Facebook controversy == Breakup Notifier gathered 100,000 users in less than 24 hours of its launch and reached a user base of more than 3,000,000 in February 2011. Facebook then blocked the app. Loewenherz later created an app named Crush Notifier, which differs from the original app in that users can check if they have a mutual crush. Breakup Notifier was later unblocked by Facebook and monetized.
Softwarp
Softwarp is a software technique to warp an image so that it can be projected on a curved screen. This can be done in real time by inserting the softwarp as a last step in the rendering cycle. The problem is to know how the image should be warped to look correct on the curved screen. There are several techniques to auto calibrate the warping by projecting a pattern and using cameras and/or sensors. The information from the sensors is sent to the software so that it can analyze the data and calculate the curvature of the projection screen. == Usage == The softwarp can be used to project virtual views on curved walls and domes. These are usually used in vehicle simulators, for instance boat-, car- and airplane simulators. To make it possible to cover a dome with a 360 degree view you need to use several projectors. A problem with using several projectors on the same screen is that the edges between the projected images get about twice the amount of light. This is solved by using a technique called edge blending. With this technique a “filter” is inserted on the edge that fades the image from 100% light strength (luminance) to 0% (the lowest luminance depends on the contrast ratio of the projector). == History == The first warping technologies used a hardware image processing unit to warp the image. This processing unit was inserted between the graphics card and the projector. The problem with this technique is that it depends on the type of signal and the quality of the signal from the graphics card to warp it correctly. The process unit also needs several lines of image information before it can start sending out the warped image. This adds a latency to the display system that could be a problem in simulators that need fast response time, for instance fighter jet simulators. Softwarping eliminates the latency.
Energy-based model
An energy-based model (EBM), also called Canonical Ensemble Learning (CEL) or Learning via Canonical Ensemble (LCE), is an application of canonical ensemble formulation from statistical physics for learning from data. The approach prominently appears in generative artificial intelligence. EBMs provide a unified framework for many probabilistic and non-probabilistic approaches to such learning, particularly for training graphical and other structured models. An EBM learns the characteristics of a target dataset and generates a similar but larger dataset. EBMs detect the latent variables of a dataset and generate new datasets with a similar distribution. Energy-based generative neural networks is a class of generative models, which aim to learn explicit probability distributions of data in the form of energy-based models, the energy functions of which are parameterized by modern deep neural networks. Boltzmann machines are a special form of energy-based models with a specific parametrization of the energy. == Description == For a given input x {\displaystyle x} , the model describes an energy E θ ( x ) {\displaystyle E_{\theta }(x)} such that the Boltzmann distribution P θ ( x ) = e − β E θ ( x ) Z ( θ ) {\displaystyle P_{\theta }(x)={e^{-\beta E_{\theta }(x)} \over Z(\theta )}} is a probability (density), and typically β = 1 {\displaystyle \beta =1} . Since the normalization constant: Z ( θ ) := ∫ x ∈ X e − β E θ ( x ) d x {\displaystyle Z(\theta ):=\int _{x\in X}e^{-\beta E_{\theta }(x)}dx} (also known as the partition function) depends on all the Boltzmann factors of all possible inputs x {\displaystyle x} , it cannot be easily computed or reliably estimated during training simply using standard maximum likelihood estimation. However, for maximizing the likelihood during training, the gradient of the log-likelihood of a single training example x {\displaystyle x} is given by using the chain rule: ∂ θ log ( P θ ( x ) ) = E x ′ ∼ P θ [ ∂ θ E θ ( x ′ ) ] − ∂ θ E θ ( x ) ( ∗ ) {\displaystyle \partial _{\theta }\log \left(P_{\theta }(x)\right)=\mathbb {E} _{x'\sim P_{\theta }}[\partial _{\theta }E_{\theta }(x')]-\partial _{\theta }E_{\theta }(x)\,()} The expectation in the above formula for the gradient can be approximately estimated by drawing samples x ′ {\displaystyle x'} from the distribution P θ {\displaystyle P_{\theta }} using Markov chain Monte Carlo (MCMC). Early energy-based models, such as the 2003 Boltzmann machine by Hinton, estimated this expectation via blocked Gibbs sampling. Newer approaches make use of more efficient Stochastic Gradient Langevin Dynamics (LD), drawing samples using: x 0 ′ ∼ P 0 , x i + 1 ′ = x i ′ − α 2 ∂ E θ ( x i ′ ) ∂ x i ′ + ϵ {\displaystyle x_{0}'\sim P_{0},x_{i+1}'=x_{i}'-{\frac {\alpha }{2}}{\frac {\partial E_{\theta }(x_{i}')}{\partial x_{i}'}}+\epsilon } , where ϵ ∼ N ( 0 , α ) {\displaystyle \epsilon \sim {\mathcal {N}}(0,\alpha )} . A replay buffer of past values x i ′ {\displaystyle x_{i}'} is used with LD to initialize the optimization module. The parameters θ {\displaystyle \theta } of the neural network are therefore trained in a generative manner via MCMC-based maximum likelihood estimation: the learning process follows an "analysis by synthesis" scheme, where within each learning iteration, the algorithm samples the synthesized examples from the current model by a gradient-based MCMC method (e.g., Langevin dynamics or Hybrid Monte Carlo), and then updates the parameters θ {\displaystyle \theta } based on the difference between the training examples and the synthesized ones – see equation ( ∗ ) {\displaystyle ()} . This process can be interpreted as an alternating mode seeking and mode shifting process, and also has an adversarial interpretation. Essentially, the model learns a function E θ {\displaystyle E_{\theta }} that associates low energies to correct values, and higher energies to incorrect values. After training, given a converged energy model E θ {\displaystyle E_{\theta }} , the Metropolis–Hastings algorithm can be used to draw new samples. The acceptance probability is given by: P a c c ( x i → x ∗ ) = min ( 1 , P θ ( x ∗ ) P θ ( x i ) ) . {\displaystyle P_{acc}(x_{i}\to x^{})=\min \left(1,{\frac {P_{\theta }(x^{})}{P_{\theta }(x_{i})}}\right).} == History == The term "energy-based models" was first coined in a 2003 JMLR paper where the authors defined a generalisation of independent components analysis to the overcomplete setting using EBMs. Other early work on EBMs proposed models that represented energy as a composition of latent and observable variables. == Characteristics == EBMs demonstrate useful properties: Simplicity and stability. The EBM is the only object that needs to be designed and trained. Separate networks need not be trained to ensure balance. Adaptive computation time. An EBM can generate sharp, diverse samples or (more quickly) coarse, less diverse samples. Given infinite time, this procedure produces true samples. Flexibility. In Variational Autoencoders (VAE) and flow-based models, the generator learns a map from a continuous space to a (possibly) discontinuous space containing different data modes. EBMs can learn to assign low energies to disjoint regions (multiple modes). Adaptive generation. EBM generators are implicitly defined by the probability distribution, and automatically adapt as the distribution changes (without training), allowing EBMs to address domains where generator training is impractical, as well as minimizing mode collapse and avoiding spurious modes from out-of-distribution samples. Compositionality. Individual models are unnormalized probability distributions, allowing models to be combined through product of experts or other hierarchical techniques. == Experimental results == On image datasets such as CIFAR-10 and ImageNet 32x32, an EBM model generated high-quality images relatively quickly. It supported combining features learned from one type of image for generating other types of images. It was able to generalize using out-of-distribution datasets, outperforming flow-based and autoregressive models. EBM was relatively resistant to adversarial perturbations, behaving better than models explicitly trained against them with training for classification. == Applications == Target applications include natural language processing, robotics and computer vision. The first energy-based generative neural network is the generative ConvNet proposed in 2016 for image patterns, where the neural network is a convolutional neural network. The model has been generalized to various domains to learn distributions of videos, and 3D voxels. They are made more effective in their variants. They have proven useful for data generation (e.g., image synthesis, video synthesis, 3D shape synthesis, etc.), data recovery (e.g., recovering videos with missing pixels or image frames, 3D super-resolution, etc), data reconstruction (e.g., image reconstruction and linear interpolation ). == Alternatives == EBMs compete with techniques such as variational autoencoders (VAEs), generative adversarial networks (GANs) or normalizing flows. == Extensions == === Joint energy-based models === Joint energy-based models (JEM), proposed in 2020 by Grathwohl et al., allow any classifier with softmax output to be interpreted as energy-based model. The key observation is that such a classifier is trained to predict the conditional probability p θ ( y | x ) = e f → θ ( x ) [ y ] ∑ j = 1 K e f → θ ( x ) [ j ] for y = 1 , … , K and f → θ = ( f 1 , … , f K ) ∈ R K , {\displaystyle p_{\theta }(y|x)={\frac {e^{{\vec {f}}_{\theta }(x)[y]}}{\sum _{j=1}^{K}e^{{\vec {f}}_{\theta }(x)[j]}}}\ \ {\text{ for }}y=1,\dotsc ,K{\text{ and }}{\vec {f}}_{\theta }=(f_{1},\dotsc ,f_{K})\in \mathbb {R} ^{K},} where f → θ ( x ) [ y ] {\displaystyle {\vec {f}}_{\theta }(x)[y]} is the y-th index of the logits f → {\displaystyle {\vec {f}}} corresponding to class y. Without any change to the logits it was proposed to reinterpret the logits to describe a joint probability density: p θ ( y , x ) = e f → θ ( x ) [ y ] Z ( θ ) , {\displaystyle p_{\theta }(y,x)={\frac {e^{{\vec {f}}_{\theta }(x)[y]}}{Z(\theta )}},} with unknown partition function Z ( θ ) {\displaystyle Z(\theta )} and energy E θ ( x , y ) = − f θ ( x ) [ y ] {\displaystyle E_{\theta }(x,y)=-f_{\theta }(x)[y]} . By marginalization, we obtain the unnormalized density p θ ( x ) = ∑ y p θ ( y , x ) = ∑ y e f → θ ( x ) [ y ] Z ( θ ) =: e − E θ ( x ) , {\displaystyle p_{\theta }(x)=\sum _{y}p_{\theta }(y,x)=\sum _{y}{\frac {e^{{\vec {f}}_{\theta }(x)[y]}}{Z(\theta )}}=:e^{-E_{\theta }(x)},} therefore, E θ ( x ) = − log ( ∑ y e f → θ ( x ) [ y ] Z ( θ ) ) , {\displaystyle E_{\theta }(x)=-\log \left(\sum _{y}{\frac {e^{{\vec {f}}_{\theta }(x)[y]}}{Z(\theta )}}\right),} so that any classifier can be used to define an energy function E θ ( x ) {\displaystyle E_{\theta }(x)} .
Stairstep interpolation
In the field of image processing, stairstep interpolation is a widely employed method technique for interpolating pixels after enlarging an image. The fundamental concept is to interpolate multiple times, in small increments, using any interpolation algorithm that is better than nearest-neighbor interpolation such as; bilinear interpolation, and bicubic interpolation. A common scenario is to interpolate an image by using a bicubic interpolation which increases the image size by no more than 10% (110% of the original size) at a time until the desired size is reached. Fred Miranda, a developer, popularized this method by creating and developing several Photoshop plug-ins that incorporate this technique. == Example ==