In graphic design and computer graphics, a drop shadow is a visual effect consisting of a drawing element which looks like the shadow of an object, giving the impression that the object is raised above the objects behind it. The drop shadow is often used for elements of a graphical user interface such as windows or menus, and for simple text. The text label for icons on desktops in many desktop environments has a drop shadow, as this effect effectively distinguishes the text from any colored background it may be in front of. A simple way of drawing a drop shadow of a rectangular object is to draw a gray or black area underneath and offset from the object. In general, a drop shadow is a copy in black or gray of the object, drawn in a slightly different position. Realism may be increased by: Darkening the colors of the pixels where the shadow casts instead of making them gray. This can be done with alpha blending the shadow with the area it is cast on. Softening the edges of the shadow. This can be done by adding Gaussian blur to the shadow's alpha channel before blending. Inset drop shadows are a type which draws the shadows inside the element. This allows the interface element to appear as if it is sunken into the interface. == Photo editing == In photo editing or photography post-production, a drop shadow may be added right beneath a model or product in the image. It is used to create contrast between the background and the subject. To add a drop shadow, retouchers use graphic editing tools like Adobe Photoshop. Drop shadows are often used as a visual effect in e-commerce. This is done to improve the presentation of product images and create depth in the image. == Use == Generally, window managers which are capable of compositing allow drop shadow effects, whereas incapable window managers do not. In some operating systems like macOS, drop shadow is used to differentiate between active and inactive windows. Websites are able to use drop shadow effects through the CSS properties box-shadow, text-shadow, and drop-shadow() filter function in filter. The first two are used for elements and text respectively, while the filter applies to the element's content, letting it support oddly shaped elements or transparent images.
Oracle Cloud Platform
Oracle Cloud Platform refers to a Platform as a Service (PaaS) offerings by Oracle Corporation as part of Oracle Cloud Infrastructure. These offerings are used to build, deploy, integrate and extend applications in the cloud. The offerings support a variety of programming languages, databases, tools and frameworks including Oracle-specific, open source and third-party software and systems. == Deployment models == Oracle Cloud Platform offers public, private and hybrid cloud deployment models. == Architecture == Oracle Cloud Platform provides both Infrastructure as a Service (IaaS) and Platform as a Service (PaaS). The infrastructure is offered through a global network of Oracle managed data centers. Oracle deploys their cloud in Regions. Inside each Region are at least three fault-independent Availability Domains. Each of these Availability Domains contains an independent data center with power, thermal and network isolation. Oracle Cloud is generally available in North America, EMEA, APAC and Japan with announced South America and US Govt. regions coming soon.
Distributional–relational database
A distributional–relational database, or word-vector database, is a database management system (DBMS) that uses distributional word-vector representations to enrich the semantics of structured data. As distributional word-vectors can be built automatically from large-scale corpora, this enrichment supports the construction of databases which can embed large-scale commonsense background knowledge into their operations. Distributional-Relational models can be applied to the construction of schema-agnostic databases (databases in which users can query the data without being aware of its schema), semantic search, schema-integration and inductive and abductive reasoning as well as different applications in which a semantically flexible knowledge representation model is needed. The main advantage of distributional–relational models over purely logical or semantic web models is the fact that the core semantic associations can be automatically captured from corpora, in contrast to the definition of manually curated ontologies and rule knowledge bases. == Distributional–relational models == Distributional–relational models were first formalized as a mechanism to cope with the vocabulary/semantic gap between users and the schema behind the data. In this scenario, distributional semantic relatedness measures, combined with semantic pivoting heuristics can support the approximation between user queries (expressed in their own vocabulary), and data (expressed in the vocabulary of the designer). In this model, the database symbols (entities and relations) are embedded into a distributional semantic space and have a geometric interpretation under a latent or explicit semantic space. The geometric aspect supports the semantic approximation between entities from different databases, or between a query term and a database entity. The distributional relational model then becomes a double layered model where the semantics of the structured data provides the fine-grained semantics intended by the database designer, which is extended by the distributional semantic model which contains the semantic associations expressed at a broader use. These models support the generalization from a closed communication scenario (in which database designers and users live in the same context, e.g. the same organization) to an open communication scenario (e.g. different organizations, the Web), creating an abstraction layer between users and the specific representation of the conceptual model.
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Oren Etzioni
Oren Etzioni (born 1964) is Professor Emeritus of Computer Science at the University of Washington, and founding CEO of the Allen Institute for Artificial Intelligence (AI2). Etzioni is a co-founder of Vercept, an AI startup, and founder and CEO of TrueMedia.org, a non-profit dedicated to fighting political deepfakes, which launched in April 2024. He is also the Founder and Technical Director of the AI2 Incubator and a venture partner at the Madrona Venture Group. == Early life and education == Etzioni is the son of Israeli-American intellectual Amitai Etzioni. He was the first student to major in computer science at Harvard University, where he earned a bachelor's degree in 1986. He earned a PhD from Carnegie Mellon University in January, 1991, supervised by Tom M. Mitchell. == University of Washington career == Etzioni joined the University of Washington faculty in 1991, immediately after receiving his PhD. He rose through the ranks to become the Washington Research Foundation Entrepreneurship Professor in Computer Science & Engineering. Etzioni's research has been focused on basic problems in the study of intelligence, machine reading, machine learning and web search. Past projects include Internet Softbots—the study of intelligent agents in the context of real-world software testbeds. In 2003, he started the KnowItAll project for acquiring massive amounts of information from the web. In 2005, he founded and became the director of the university's Turing Center. The center investigated problems in data mining, natural language processing, the Semantic Web and other web search topics. Etzioni coined the term machine reading and helped to create the first commercial comparison shopping agent. He has published over 200 technical papers, and his H-index exceeds 100. == Entrepreneurship == As a faculty member Etzioni was also an active entrepreneur, founding multiple companies and pioneering multiple technologies including MetaCrawler (bought by Infospace), Netbot (bought by Excite in 1997 for $35 million), and ClearForest (bought by Reuters). He founded Farecast, a travel metasearch and price prediction site, which was acquired by Microsoft in 2008 for $115 million. Before founding Farecast, he developed a program originally called Hamlet, that used algorithms to identify patterns in airfare data using data-mining techniques. He also co-founded Decide.com, a website to help consumers make buying decisions using previous price history and recommendations from other users. Decide.com was bought by eBay in September, 2013. Etzioni is also a venture partner at the Madrona Venture Group. He is founder and CEO of TrueMedia.org, a non-profit dedicated to fighting political deepfakes, which launched in April 2024. Etzioni is a co-founder of Vercept, an AI startup formed in 2025. == Founding CEO of AI2 == In September 2013 Etzioni was selected as the Founding CEO of the Allen Institute for Artificial Intelligence by philanthropist Paul G. Allen, and in January 2014 he took a leave of absence from the University of Washington to serve in that role. Etzioni's technical contributions continued at AI2; for example, in 2015, he helped to create the Semantic Scholar search engine. Under Etzioni’s leadership, AI2 grew from zero to over two hundred team members including notable researchers and engineers across several domains of AI. By 2021, its AI2 researchers had published near 700 papers in publications such as AAAI, ACL, CVPR, NeurIPS, and ICLR. Twenty-four of these papers had garnered special-recognition awards. AI2 also offered several key resources and tools to the AI community including the AllenNLP library, Semantic Scholar, and the conservation platforms EarthRanger and Skylight. Ed Lazowska, AI2 Board Member, has stated about Etzioni that he "took the collegial, collaborative culture that he absorbed in his 20+ years as a professor in UW's Allen School and mixed it with the singular focus that drives startups to create an elixir that AI2 folks have been drinking over the last eight years. The result is an exceptional organization of scientists, engineers, and entrepreneurs that's pursuing Paul Allen’s vision of ‘AI for the Common Good’ with extraordinary success.” == Popular press == In addition to his scientific publications, Etzioni has written commentary on AI for The New York Times, Wired, Nature, and other publications. After reading the idea in a book about AI by Brad Smith and Harry Shum, Etzioni has attempted to create an oath for AI practitioners. In 2018, he published what he called a "Hippocratic Oath for artificial intelligence practitioners" in TechCrunch. == Awards and recognition == In 1993, Etzioni received a National Young Investigator Award. In 2003, Etzioni was elected as AAAI Fellow. In 2005, Etzioni received an IJCAI Distinguished Paper Award for "A Probabilistic Model of Redundancy in Information Extraction". In 2007, he received the Robert S. Engelmore Memorial Award. In 2012 Etzioni was featured as GeekWire's "Geek of the Week". In 2013 Etzioni was voted "Geek of the Year" through GeekWire. In 2022, Etzioni received the 2012 ACL Test-of-Time Paper Award. In 2022, Etzioni, along with Ana-Maria Popescu and Henry Kautz, received the ACM Intelligent User Interfaces Most Impact Award for their 2003 paper, "Towards a Theory of Natural Language Interfaces to Databases". == Personal life == Etzioni has three children, and has said in interviews that family is his number one priority. He is married to Ivone Etzioni, and was previously married to Dr. Ruth Etzioni, a biostatistician at the Fred Hutchinson Cancer Center. Outside of his professional career, Etzioni has a wide range of personal interests. He has attended the Burning Man festival, which he described as a valuable way to step outside his comfort zone. His first computer was a TRS-80, and he has described his car’s GPS as his favorite gadget, joking that he has “no sense of direction.” == Selected publications == === Scholarly publications === Etzioni, Oren (July 1994). "A Softbot-based Interface to the Internet" (PDF). Communications of the ACM. Retrieved March 29, 2018. Etzioni, Oren (December 2008). "Open Information Extraction from the Web" (PDF). Communications of the ACM. Retrieved March 29, 2018. Zamir, Oren; Etzioni, Oren (1998). "Web document clustering". Proceedings of the 21st annual international ACM SIGIR conference on Research and development in information retrieval. ACM. pp. 46–54. doi:10.1145/290941.290956. ISBN 978-1-58113-015-7. S2CID 244069. Zamir, Oren; Etzioni, Oren (May 1999). "Grouper: a dynamic clustering interface to Web search results". Computer Networks. 31 (11–16): 1361–1374. CiteSeerX 10.1.1.31.8216. doi:10.1016/S1389-1286(99)00054-7. S2CID 206134308. Popescu, Ana-Maria; Etzioni, Oren (2005). "Extracting product features and opinions from reviews". Proceedings of the conference on Human Language Technology and Empirical Methods in Natural Language Processing - HLT '05. pp. 339–346. doi:10.3115/1220575.1220618. Etzioni, Oren; Cafarella, Michael; Downey, Doug; Popescu, Ana-Maria; Shaked, Tal; Sonderland, Stephen; Weld, Daniel; Yates, Alexander (June 2005). "Unsupervised named-entity extraction from the Web: An experimental study". Artificial Intelligence. 165 (1): 91–134. doi:10.1016/j.artint.2005.03.001. Downey, Doug; Etzioni, Oren; Sonderland, Stephen (July 2010). "Grouper: Analysis of a probabilistic model of redundancy in unsupervised information extraction". Artificial Intelligence. 174 (11): 726–748. CiteSeerX 10.1.1.174.2441. doi:10.1016/j.artint.2010.04.024. === Popular articles === Etzioni, Oren (August 4, 2011). "Web Search Needs a Shakeup" (PDF). Nature. Retrieved November 21, 2019. Etzioni, Oren (December 9, 2014). "AI Won't Exterminate Us – It Will Empower Us". Backchannel. Retrieved March 29, 2018. Etzioni, Oren (February 4, 2016). "To Keep AI Safe -- Use AI". Vox. Retrieved November 21, 2019. Etzioni, Oren (April 8, 2016). "Quora Session with Oren Etzioni". Quora. Retrieved March 29, 2018. Etzioni, Oren (June 15, 2016). "Deep Learning Isn't a Dangerous Magic Genie. It's Just Math". Wired. Retrieved March 29, 2018. Etzioni, Oren (September 20, 2016). "No, the Experts Don't Think Superintelligent AI is a Threat to Humanity". MIT Technology Review. Retrieved November 21, 2019. Etzioni, Oren (July 6, 2017). "Artificial intelligence: AI Zooms in on highly influential citations". Nature. Retrieved March 29, 2018. Etzioni, Oren (September 1, 2017). "How to Regulate Artificial Intelligence". The New York Times. Retrieved March 29, 2018. Etzioni, Oren (November 2, 2017). "Workers Displaced by Automation Should Try A New Job: Caregiver". Wired. Retrieved March 29, 2018. Etzioni, Oren (March 14, 2018). "A Hippocratic Oath for artificial intelligence practitioners". Tech Crunch. Retrieved March 29, 2018. Etzioni, Oren (March 7, 2018). "A 'Manhattan Project' for science research". The Hill. Retrieved November 21, 2019. Etzioni, Ore
Progressive Graphics File
PGF (Progressive Graphics File) is a wavelet-based bitmapped image format that employs lossless and lossy data compression. PGF was created to improve upon and replace the JPEG format. It was developed at the same time as JPEG 2000 but with a focus on speed over compression ratio. PGF can operate at higher compression ratios without taking more encoding/decoding time and without generating the characteristic "blocky and blurry" artifacts of the original DCT-based JPEG standard. It also allows more sophisticated progressive downloads. == Color models == PGF supports a wide variety of color models: Grayscale with 1, 8, 16, or 31 bits per pixel Indexed color with palette size of 256 RGB color image with 12, 16 (red: 5 bits, green: 6 bits, blue: 5 bits), 24, or 48 bits per pixel ARGB color image with 32 bits per pixel Lab color image with 24 or 48 bits per pixel CMYK color image with 32 or 64 bits per pixel == Technical discussion == PGF claims to achieve an improved compression quality over JPEG adding or improving features such as scalability. Its compression performance is similar to the original JPEG standard. Very low and very high compression rates (including lossless compression) are also supported in PGF. The ability of the design to handle a very large range of effective bit rates is one of the strengths of PGF. For example, to reduce the number of bits for a picture below a certain amount, the advisable thing to do with the first JPEG standard is to reduce the resolution of the input image before encoding it — something that is ordinarily not necessary for that purpose when using PGF because of its wavelet scalability properties. The PGF process chain contains the following four steps: Color space transform (in case of color images) Discrete Wavelet Transform Quantization (in case of lossy data compression) Hierarchical bit-plane run-length encoding === Color components transformation === Initially, images have to be transformed from the RGB color space to another color space, leading to three components that are handled separately. PGF uses a fully reversible modified YUV color transform. The transformation matrices are: [ Y r U r V r ] = [ 1 4 1 2 1 4 1 − 1 0 0 − 1 1 ] [ R G B ] ; [ R G B ] = [ 1 3 4 − 1 4 1 − 1 4 − 1 4 1 − 1 4 3 4 ] [ Y r U r V r ] {\displaystyle {\begin{bmatrix}Y_{r}\\U_{r}\\V_{r}\end{bmatrix}}={\begin{bmatrix}{\frac {1}{4}}&{\frac {1}{2}}&{\frac {1}{4}}\\1&-1&0\\0&-1&1\end{bmatrix}}{\begin{bmatrix}R\\G\\B\end{bmatrix}};\qquad \qquad {\begin{bmatrix}R\\G\\B\end{bmatrix}}={\begin{bmatrix}1&{\frac {3}{4}}&-{\frac {1}{4}}\\1&-{\frac {1}{4}}&-{\frac {1}{4}}\\1&-{\frac {1}{4}}&{\frac {3}{4}}\end{bmatrix}}{\begin{bmatrix}Y_{r}\\U_{r}\\V_{r}\end{bmatrix}}} The chrominance components can be, but do not necessarily have to be, down-scaled in resolution. === Wavelet transform === The color components are then wavelet transformed to an arbitrary depth. In contrast to JPEG 1992 which uses an 8x8 block-size discrete cosine transform, PGF uses one reversible wavelet transform: a rounded version of the biorthogonal CDF 5/3 wavelet transform. This wavelet filter bank is exactly the same as the reversible wavelet used in JPEG 2000. It uses only integer coefficients, so the output does not require rounding (quantization) and so it does not introduce any quantization noise. === Quantization === After the wavelet transform, the coefficients are scalar-quantized to reduce the amount of bits to represent them, at the expense of a loss of quality. The output is a set of integer numbers which have to be encoded bit-by-bit. The parameter that can be changed to set the final quality is the quantization step: the greater the step, the greater is the compression and the loss of quality. With a quantization step that equals 1, no quantization is performed (it is used in lossless compression). In contrast to JPEG 2000, PGF uses only powers of two, therefore the parameter value i represents a quantization step of 2i. Just using powers of two makes no need of integer multiplication and division operations. === Coding === The result of the previous process is a collection of sub-bands which represent several approximation scales. A sub-band is a set of coefficients — integer numbers which represent aspects of the image associated with a certain frequency range as well as a spatial area of the image. The quantized sub-bands are split further into blocks, rectangular regions in the wavelet domain. They are typically selected in a way that the coefficients within them across the sub-bands form approximately spatial blocks in the (reconstructed) image domain and collected in a fixed size macroblock. The encoder has to encode the bits of all quantized coefficients of a macroblock, starting with the most significant bits and progressing to less significant bits. In this encoding process, each bit-plane of the macroblock gets encoded in two so-called coding passes, first encoding bits of significant coefficients, then refinement bits of significant coefficients. Clearly, in lossless mode all bit-planes have to be encoded, and no bit-planes can be dropped. Only significant coefficients are compressed with an adaptive run-length/Rice (RLR) coder, because they contain long runs of zeros. The RLR coder with parameter k (logarithmic length of a run of zeros) is also known as the elementary Golomb code of order 2k. === Comparison with other file formats === JPEG 2000 is slightly more space-efficient in handling natural images. The PSNR for the same compression ratio is on average 3% better than the PSNR of PGF. It has a small advantage in compression ratio but longer encoding and decoding times. PNG (Portable Network Graphics) is more space-efficient in handling images with many pixels of the same color. There are several self-proclaimed advantages of PGF over the ordinary JPEG standard: Superior compression performance: The image quality (measured in PSNR) for the same compression ratio is on average 3% better than the PSNR of JPEG. At lower bit rates (e.g. less than 0.25 bits/pixel for gray-scale images), PGF has a much more significant advantage over certain modes of JPEG: artifacts are less visible and there is almost no blocking. The compression gains over JPEG are attributed to the use of DWT. Multiple resolution representation: PGF provides seamless compression of multiple image components, with each component carrying from 1 to 31 bits per component sample. With this feature there is no need for separately stored preview images (thumbnails). Progressive transmission by resolution accuracy, commonly referred to as progressive decoding: PGF provides efficient code-stream organizations which are progressive by resolution. This way, after a smaller part of the whole file has been received, it is possible to see a lower quality of the final picture, the quality can be improved monotonically getting more data from the source. Lossless and lossy compression: PGF provides both lossless and lossy compression in a single compression architecture. Both lossy and lossless compression are provided by the use of a reversible (integer) wavelet transform. Side channel spatial information: Transparency and alpha planes are fully supported ROI extraction: Since version 5, PGF supports extraction of regions of interest (ROI) without decoding the whole image. == Available software == The author published libPGF via a SourceForge, under the GNU Lesser General Public License version 2.0. Xeraina offers a free Windows console encoder and decoder, and PGF viewers based on WIC for 32bit and 64bit Windows platforms. Other WIC applications including File Explorer are able to display PGF images after installing this viewer. Digikam is a popular open-source image editing and cataloging software that uses libPGF for its thumbnails. It makes use of the progressive decoding feature of PGF images to store a single version of each thumbnail, which can then be decoded to different resolutions without loss, thus allowing users to dynamically change the size of the thumbnails without having to recalculate them again.
Wasserstein GAN
The Wasserstein Generative Adversarial Network (WGAN) is a variant of generative adversarial network (GAN) proposed in 2017 that aims to "improve the stability of learning, get rid of problems like mode collapse, and provide meaningful learning curves useful for debugging and hyperparameter searches". Compared with the original GAN discriminator, the Wasserstein GAN discriminator provides a better learning signal to the generator. This allows the training to be more stable when generator is learning distributions in very high dimensional spaces. == Motivation == === The GAN game === The original GAN method is based on the GAN game, a zero-sum game with 2 players: generator and discriminator. The game is defined over a probability space ( Ω , B , μ r e f ) {\displaystyle (\Omega ,{\mathcal {B}},\mu _{ref})} , The generator's strategy set is the set of all probability measures μ G {\displaystyle \mu _{G}} on ( Ω , B ) {\displaystyle (\Omega ,{\mathcal {B}})} , and the discriminator's strategy set is the set of measurable functions D : Ω → [ 0 , 1 ] {\displaystyle D:\Omega \to [0,1]} . The objective of the game is L ( μ G , D ) := E x ∼ μ r e f [ ln D ( x ) ] + E x ∼ μ G [ ln ( 1 − D ( x ) ) ] . {\displaystyle L(\mu _{G},D):=\mathbb {E} _{x\sim \mu _{ref}}[\ln D(x)]+\mathbb {E} _{x\sim \mu _{G}}[\ln(1-D(x))].} The generator aims to minimize it, and the discriminator aims to maximize it. A basic theorem of the GAN game states that Repeat the GAN game many times, each time with the generator moving first, and the discriminator moving second. Each time the generator μ G {\displaystyle \mu _{G}} changes, the discriminator must adapt by approaching the ideal D ∗ ( x ) = d μ r e f d ( μ r e f + μ G ) . {\displaystyle D^{}(x)={\frac {d\mu _{ref}}{d(\mu _{ref}+\mu _{G})}}.} Since we are really interested in μ r e f {\displaystyle \mu _{ref}} , the discriminator function D {\displaystyle D} is by itself rather uninteresting. It merely keeps track of the likelihood ratio between the generator distribution and the reference distribution. At equilibrium, the discriminator is just outputting 1 2 {\displaystyle {\frac {1}{2}}} constantly, having given up trying to perceive any difference. Concretely, in the GAN game, let us fix a generator μ G {\displaystyle \mu _{G}} , and improve the discriminator step-by-step, with μ D , t {\displaystyle \mu _{D,t}} being the discriminator at step t {\displaystyle t} . Then we (ideally) have L ( μ G , μ D , 1 ) ≤ L ( μ G , μ D , 2 ) ≤ ⋯ ≤ max μ D L ( μ G , μ D ) = 2 D J S ( μ r e f ‖ μ G ) − 2 ln 2 , {\displaystyle L(\mu _{G},\mu _{D,1})\leq L(\mu _{G},\mu _{D,2})\leq \cdots \leq \max _{\mu _{D}}L(\mu _{G},\mu _{D})=2D_{JS}(\mu _{ref}\|\mu _{G})-2\ln 2,} so we see that the discriminator is actually lower-bounding D J S ( μ r e f ‖ μ G ) {\displaystyle D_{JS}(\mu _{ref}\|\mu _{G})} . === Wasserstein distance === Thus, we see that the point of the discriminator is mainly as a critic to provide feedback for the generator, about "how far it is from perfection", where "far" is defined as Jensen–Shannon divergence. Naturally, this brings the possibility of using a different criteria of farness. There are many possible divergences to choose from, such as the f-divergence family, which would give the f-GAN. The Wasserstein GAN is obtained by using the Wasserstein metric, which satisfies a "dual representation theorem" that renders it highly efficient to compute: A proof can be found in the main page on Wasserstein metric. == Definition == By the Kantorovich-Rubenstein duality, the definition of Wasserstein GAN is clear:A Wasserstein GAN game is defined by a probability space ( Ω , B , μ r e f ) {\displaystyle (\Omega ,{\mathcal {B}},\mu _{ref})} , where Ω {\displaystyle \Omega } is a metric space, and a constant K > 0 {\displaystyle K>0} . There are 2 players: generator and discriminator (also called "critic"). The generator's strategy set is the set of all probability measures μ G {\displaystyle \mu _{G}} on ( Ω , B ) {\displaystyle (\Omega ,{\mathcal {B}})} . The discriminator's strategy set is the set of measurable functions of type D : Ω → R {\displaystyle D:\Omega \to \mathbb {R} } with bounded Lipschitz-norm: ‖ D ‖ L ≤ K {\displaystyle \|D\|_{L}\leq K} . The Wasserstein GAN game is a zero-sum game, with objective function L W G A N ( μ G , D ) := E x ∼ μ G [ D ( x ) ] − E x ∼ μ r e f [ D ( x ) ] . {\displaystyle L_{WGAN}(\mu _{G},D):=\mathbb {E} _{x\sim \mu _{G}}[D(x)]-\mathbb {E} _{x\sim \mu _{ref}}[D(x)].} The generator goes first, and the discriminator goes second. The generator aims to minimize the objective, and the discriminator aims to maximize the objective: min μ G max D L W G A N ( μ G , D ) . {\displaystyle \min _{\mu _{G}}\max _{D}L_{WGAN}(\mu _{G},D).} By the Kantorovich-Rubenstein duality, for any generator strategy μ G {\displaystyle \mu _{G}} , the optimal reply by the discriminator is D ∗ {\displaystyle D^{}} , such that L W G A N ( μ G , D ∗ ) = K ⋅ W 1 ( μ G , μ r e f ) . {\displaystyle L_{WGAN}(\mu _{G},D^{})=K\cdot W_{1}(\mu _{G},\mu _{ref}).} Consequently, if the discriminator is good, the generator would be constantly pushed to minimize W 1 ( μ G , μ r e f ) {\displaystyle W_{1}(\mu _{G},\mu _{ref})} , and the optimal strategy for the generator is just μ G = μ r e f {\displaystyle \mu _{G}=\mu _{ref}} , as it should. == Comparison with GAN == In the Wasserstein GAN game, the discriminator provides a better gradient than in the GAN game. Consider for example a game on the real line where both μ G {\displaystyle \mu _{G}} and μ r e f {\displaystyle \mu _{ref}} are Gaussian. Then the optimal Wasserstein critic D W G A N {\displaystyle D_{WGAN}} and the optimal GAN discriminator D {\displaystyle D} are plotted as below: For fixed discriminator, the generator needs to minimize the following objectives: For GAN, E x ∼ μ G [ ln ( 1 − D ( x ) ) ] {\displaystyle \mathbb {E} _{x\sim \mu _{G}}[\ln(1-D(x))]} . For Wasserstein GAN, E x ∼ μ G [ D W G A N ( x ) ] {\displaystyle \mathbb {E} _{x\sim \mu _{G}}[D_{WGAN}(x)]} . Let μ G {\displaystyle \mu _{G}} be parametrized by θ {\displaystyle \theta } , then we can perform stochastic gradient descent by using two unbiased estimators of the gradient: ∇ θ E x ∼ μ G [ ln ( 1 − D ( x ) ) ] = E x ∼ μ G [ ln ( 1 − D ( x ) ) ⋅ ∇ θ ln ρ μ G ( x ) ] {\displaystyle \nabla _{\theta }\mathbb {E} _{x\sim \mu _{G}}[\ln(1-D(x))]=\mathbb {E} _{x\sim \mu _{G}}[\ln(1-D(x))\cdot \nabla _{\theta }\ln \rho _{\mu _{G}}(x)]} ∇ θ E x ∼ μ G [ D W G A N ( x ) ] = E x ∼ μ G [ D W G A N ( x ) ⋅ ∇ θ ln ρ μ G ( x ) ] {\displaystyle \nabla _{\theta }\mathbb {E} _{x\sim \mu _{G}}[D_{WGAN}(x)]=\mathbb {E} _{x\sim \mu _{G}}[D_{WGAN}(x)\cdot \nabla _{\theta }\ln \rho _{\mu _{G}}(x)]} where we used the reparameterization trick. As shown, the generator in GAN is motivated to let its μ G {\displaystyle \mu _{G}} "slide down the peak" of ln ( 1 − D ( x ) ) {\displaystyle \ln(1-D(x))} . Similarly for the generator in Wasserstein GAN. For Wasserstein GAN, D W G A N {\displaystyle D_{WGAN}} has gradient 1 almost everywhere, while for GAN, ln ( 1 − D ) {\displaystyle \ln(1-D)} has flat gradient in the middle, and steep gradient elsewhere. As a result, the variance for the estimator in GAN is usually much larger than that in Wasserstein GAN. See also Figure 3 of. The problem with D J S {\displaystyle D_{JS}} is much more severe in actual machine learning situations. Consider training a GAN to generate ImageNet, a collection of photos of size 256-by-256. The space of all such photos is R 256 2 {\displaystyle \mathbb {R} ^{256^{2}}} , and the distribution of ImageNet pictures, μ r e f {\displaystyle \mu _{ref}} , concentrates on a manifold of much lower dimension in it. Consequently, any generator strategy μ G {\displaystyle \mu _{G}} would almost surely be entirely disjoint from μ r e f {\displaystyle \mu _{ref}} , making D J S ( μ G ‖ μ r e f ) = + ∞ {\displaystyle D_{JS}(\mu _{G}\|\mu _{ref})=+\infty } . Thus, a good discriminator can almost perfectly distinguish μ r e f {\displaystyle \mu _{ref}} from μ G {\displaystyle \mu _{G}} , as well as any μ G ′ {\displaystyle \mu _{G}'} close to μ G {\displaystyle \mu _{G}} . Thus, the gradient ∇ μ G L ( μ G , D ) ≈ 0 {\displaystyle \nabla _{\mu _{G}}L(\mu _{G},D)\approx 0} , creating no learning signal for the generator. Detailed theorems can be found in. == Training Wasserstein GANs == Training the generator in Wasserstein GAN is just gradient descent, the same as in GAN (or most deep learning methods), but training the discriminator is different, as the discriminator is now restricted to have bounded Lipschitz norm. There are several methods for this. === Upper-bounding the Lipschitz norm === Let the discriminator function D {\displaystyle D} to be implemented by a multilayer perceptron: D = D n ∘ D n − 1 ∘ ⋯ ∘ D 1 {\displaystyle D=D_{n}\circ D_{n-1}\circ \cdots \circ D_{1}} where D i ( x ) = h ( W i x ) {\displaystyle D_{i}(x)=h(W_