In mathematics, the intrinsic dimension of a subset can be thought of as the minimal number of variables needed to represent the subset. The concept has widespread applications in geometry, dynamical systems, signal processing, statistics, and other fields. Due to its widespread applications and vague conceptualization, there are many different ways to define it rigorously. Consequently, the same set might have different intrinsic dimensions according to different definitions. The intrinsic dimension can be used as a lower bound of what dimension it is possible to compress a data set into through dimension reduction, but it can also be used as a measure of the complexity of the data set or signal. For a data set or signal of N variables, its intrinsic dimension M satisfies 0 ≤ M ≤ N, although estimators may yield higher values. == Exact dimension == === Differential === In differential geometry, given a differentiable manifold N and a submanifold M, the intrinsic dimension of M is its dimension. Suppose N has n dimensions and M has m dimensions, then that means around any point in M, there exists a local coordinate system ( x 1 , … , x m , x m + 1 , … , x n ) {\displaystyle (x_{1},\dots ,x_{m},x_{m+1},\dots ,x_{n})} of N, such that the manifold M is simply the subset of N defined by x m + 1 = 0 , … , x n = 0 {\displaystyle x_{m+1}=0,\dots ,x_{n}=0} . === Metric === Given a mere metric space, we can still define its intrinsic dimension. The most general case is the Hausdorff dimension, though for metric spaces occurring in practice, the box-counting dimension and the packing dimension often are identical to the Hausdorff dimension. Let X , d {\textstyle X,d} be a metric space and A ⊂ X {\textstyle A\subset X} be totally bounded. Define the covering number N ( A , ε ) = min { k : A ⊂ ⋃ i = 1 k B ( x i , ε ) } . {\displaystyle N(A,\varepsilon )=\min \left\{k:A\subset \bigcup _{i=1}^{k}B\left(x_{i},\varepsilon \right)\right\}.} The metric entropy is H ( A , ε ) = log N ( A , ε ) {\textstyle H(A,\varepsilon )=\log N(A,\varepsilon )} (any log base). The upper and lower metric entropy dimensions are dim ¯ E A = lim sup ε ↓ 0 H ( A , ε ) log ( 1 / ε ) , dim _ E A = lim inf ε ↓ 0 H ( A , ε ) log ( 1 / ε ) . {\displaystyle {\overline {\dim }}_{E}A=\limsup _{\varepsilon \downarrow 0}{\frac {H(A,\varepsilon )}{\log(1/\varepsilon )}},\quad {\underline {\dim }}_{E}A=\liminf _{\varepsilon \downarrow 0}{\frac {H(A,\varepsilon )}{\log(1/\varepsilon )}}.} If they are equal, then dim E A {\textstyle \operatorname {dim} _{E}A} is that common value, called the metric entropy dimension. The entropy dimensions are usually used in information theory, and especially coding theory, since entropy is involved in its definition. === Topological === If X {\displaystyle X} is merely a topological space, then we can still define its intrinsic dimension, using the topological dimension or Lebesgue covering dimension. An open cover of a topological space X is a family of open sets Uα such that their union is the whole space, ∪ α {\displaystyle \cup _{\alpha }} Uα = X. The order or ply of an open cover A {\displaystyle {\mathfrak {A}}} = {Uα} is the smallest number m (if it exists) for which each point of the space belongs to at most m open sets in the cover: in other words Uα1 ∩ ⋅⋅⋅ ∩ Uαm+1 = ∅ {\displaystyle \emptyset } for α1, ..., αm+1 distinct. A refinement of an open cover A {\displaystyle {\mathfrak {A}}} = {Uα} is another open cover B {\displaystyle {\mathfrak {B}}} = {Vβ}, such that each Vβ is contained in some Uα. The covering dimension of a topological space X is defined to be the minimum value of n such that every finite open cover A {\displaystyle {\mathfrak {A}}} of X has an open refinement B {\displaystyle {\mathfrak {B}}} with order n + 1. The refinement B {\displaystyle {\mathfrak {B}}} can always be chosen to be finite. Thus, if n is finite, Vβ1 ∩ ⋅⋅⋅ ∩ Vβn+2 = ∅ {\displaystyle \emptyset } for β1, ..., βn+2 distinct. If no such minimal n exists, the space is said to have infinite covering dimension. == Introductory example == Let f ( x 1 , x 2 ) {\textstyle f(x_{1},x_{2})} be a two-variable function (or signal) which is of the form f ( x 1 , x 2 ) = g ( x 1 ) {\textstyle f(x_{1},x_{2})=g(x_{1})} for some one-variable function g which is not constant. This means that f varies, in accordance to g, with the first variable or along the first coordinate. On the other hand, f is constant with respect to the second variable or along the second coordinate. It is only necessary to know the value of one, namely the first, variable in order to determine the value of f. Hence, it is a two-variable function but its intrinsic dimension is one. A slightly more complicated example is f ( x 1 , x 2 ) = g ( x 1 + x 2 ) {\textstyle f(x_{1},x_{2})=g(x_{1}+x_{2})} . f is still intrinsic one-dimensional, which can be seen by making a variable transformation y 1 = x 1 + x 2 {\textstyle y_{1}=x_{1}+x_{2}} and y 2 = x 1 − x 2 {\textstyle y_{2}=x_{1}-x_{2}} which gives f ( y 1 + y 2 2 , y 1 − y 2 2 ) = g ( y 1 ) {\textstyle f\left({\frac {y_{1}+y_{2}}{2}},{\frac {y_{1}-y_{2}}{2}}\right)=g\left(y_{1}\right)} . Since the variation in f can be described by the single variable y1 its intrinsic dimension is one. For the case that f is constant, its intrinsic dimension is zero since no variable is needed to describe variation. For the general case, when the intrinsic dimension of the two-variable function f is neither zero or one, it is two. In the literature, functions which are of intrinsic dimension zero, one, or two are sometimes referred to as i0D, i1D or i2D, respectively. == Signal processing == In signal processing of multidimensional signals, the intrinsic dimension of the signal describes how many variables are needed to generate a good approximation of the signal. For an N-variable function f, the set of variables can be represented as an N-dimensional vector x: f = f ( x ) where x = ( x 1 , … , x N ) {\textstyle f=f\left(\mathbf {x} \right){\text{ where }}\mathbf {x} =\left(x_{1},\dots ,x_{N}\right)} . If for some M-variable function g and M × N matrix A it is the case that for all x; f ( x ) = g ( A x ) , {\textstyle f(\mathbf {x} )=g(\mathbf {Ax} ),} M is the smallest number for which the above relation between f and g can be found, then the intrinsic dimension of f is M. The intrinsic dimension is a characterization of f, it is not an unambiguous characterization of g nor of A. That is, if the above relation is satisfied for some f, g, and A, it must also be satisfied for the same f and g′ and A′ given by g ′ ( y ) = g ( B y ) {\textstyle g'\left(\mathbf {y} \right)=g\left(\mathbf {By} \right)} and A ′ = B − 1 A {\textstyle \mathbf {A'} =\mathbf {B} ^{-1}\mathbf {A} } where B is a non-singular M × M matrix, since f ( x ) = g ′ ( A ′ x ) = g ( B A ′ x ) = g ( A x ) {\textstyle f\left(\mathbf {x} \right)=g'\left(\mathbf {A'x} \right)=g\left(\mathbf {BA'x} \right)=g\left(\mathbf {Ax} \right)} . == The Fourier transform of signals of low intrinsic dimension == An N variable function which has intrinsic dimension M < N has a characteristic Fourier transform. Intuitively, since this type of function is constant along one or several dimensions its Fourier transform must appear like an impulse (the Fourier transform of a constant) along the same dimension in the frequency domain. === A simple example === Let f be a two-variable function which is i1D. This means that there exists a normalized vector n ∈ R 2 {\textstyle \mathbf {n} \in \mathbb {R} ^{2}} and a one-variable function g such that f ( x ) = g ( n T x ) {\textstyle f(\mathbf {x} )=g(\mathbf {n} ^{\operatorname {T} }\mathbf {x} )} for all x ∈ R 2 {\textstyle \mathbf {x} \in \mathbb {R} ^{2}} . If F is the Fourier transform of f (both are two-variable functions) it must be the case that F ( u ) = G ( n T u ) ⋅ δ ( m T u ) {\textstyle F\left(\mathbf {u} \right)=G\left(\mathbf {n} ^{\mathrm {T} }\mathbf {u} \right)\cdot \delta \left(\mathbf {m} ^{\mathrm {T} }\mathbf {u} \right)} . Here G is the Fourier transform of g (both are one-variable functions), δ is the Dirac impulse function and m is a normalized vector in R 2 {\textstyle \mathbb {R} ^{2}} perpendicular to n. This means that F vanishes everywhere except on a line which passes through the origin of the frequency domain and is parallel to m. Along this line F varies according to G. === The general case === Let f be an N-variable function which has intrinsic dimension M, that is, there exists an M-variable function g and M × N matrix A such that f ( x ) = g ( A x ) ∀ x {\textstyle f(\mathbf {x} )=g(\mathbf {Ax} )\quad \forall \mathbf {x} } . Its Fourier transform F can then be described as follows: F vanishes everywhere except for a subspace of dimension M The subspace M is spanned by the rows of the matrix A In the subspace, F varies according to G the Fourier transform of g == Generalizations == The type of intrinsic dimension described above assume
Genigraphics
Genigraphics is a large-format printing service bureau specializing in providing poster session services to medical and scientific conferences throughout the US and Canada. The company began in 1973 as a division of General Electric. == History == Genigraphics began as a computer graphics system, developed by General Electric in the late 1960s, for NASA to use in space flight simulation. The technologies thus developed provided a foundation for the company's expansion into the commercial market. The Computed Images System & Services division (CISS, to become Genigraphics Corporation) of GE delivered the first presentation graphics system to Amoco Oil's corporate headquarters in 1973. It was named the 100 Series, and was based on DEC's PDP 11 series of mini computer systems. The first Genigraphics systems (100 Series and 100A Series) used an array of buttons, dials, knobs and joysticks, along with a built in keyboard, as the means of user interface. The PDP-11/40 computer was housed in a tall cabinet and used random access magnetic tape drives (DECtape) for storing completed presentations. The graphics generator (Forox recorder) was capable of outputting 2,000 line resolution, suitable for 35mm and 72mm film and large sheet film positive using larger cassettes for recording. 4000 and 8000 line resolution was later achieved with duplex scanning and 4x scanning by modifying to the Forox recorder's settings menu. Subsequent models (100B,C,D,D+ and D+/GVP) replaced the knobs and dials with an on screen, text based menu system, a graphics tablet and a pen. The pen/tablet combination gave way to a mouse like device in later models, and served to provide the interface with the graphics tools. User interaction with the computer for functions such as media initialization or modem to modem data transfer required a DECwriter serial terminal. In 1982, GE divested the Genigraphics division along with a host of other "non essential" business units (Genitext, Geniponics) and Genigraphics Corporation was born. Shortly after the divestiture, the headquarters of Genigraphics was moved from Liverpool, New York to Saddle Brook, New Jersey. Major success followed as the company grew exponentially over the next few years selling both systems and slide creation services. Genigraphics film recorders produced high-resolution digital images on 35mm film. The computer-generated scenes for The Last Starfighter were calculated on a Cray X-MP supercomputer and mastered with a Genigraphics film recorder. At its peak, Genigraphics Corporation employed roughly 300 people in 24 offices worldwide, with revenues upwards of $70 million annually. By the late 1980s Genigraphics saw demand for its proprietary systems dwindle and began selling the MASTERPIECE 8770 film recorder and GRAFTIME software as a peripheral for DEC Vaxes, IBM PC AT’s, and Mac NuBus machines. But the MASTERPIECE film recorder proved too expensive to sell in volume. In 1988, the company began a partnership with Microsoft to help develop the PowerPoint software. In exchange, every copy of PowerPoint included a “Send to Genigraphics” link to have files sent to a Genigraphics service bureau to be produced as 35mm slides. This partnership continued until 2001. In 1989, after three years of flat revenue, Genigraphics sold its hardware business in order to focus on its service bureau business and partnership with Microsoft via PowerPoint. In 1994, all assets of Genigraphics, including equipment, software development, in-house artwork, trademarks, and rights to the Microsoft partnership, were sold to InFocus Corporation of Wilsonville, Oregon who continued to operate under the Genigraphics brand name. The twenty-four service bureaus were consolidated to a 20,000 square foot facility next to the FedEx hub in Memphis, Tennessee. This allowed PowerPoint slide orders to be received until 10pm and delivered across the United States by the following morning. In 1995, InFocus registered www.genigraphics.com and was among the first to offer a form of ecommerce allowing 35mm slides, color prints and transparencies, printed booklets, and digital projectors to be purchased online. In 1998, then current management bought Genigraphics from InFocus and have operated it continuously ever since as Genigraphics LLC. That same year, InFocus projector rentals were added to the “Send to Genigraphics” link in PowerPoint and Genigraphics became the rental and repair center for all InFocus national accounts. It also marked Genigraphics entry into the new industry of large format printing; leveraging their knowledge of, and access to, PowerPoint programming code to develop a proprietary printer driver to output directly to an Epson 9500 wide format printer. At the time, Genigraphics was the exclusive 35mm slide vendor for all Kinko’s stores in the United States and poster printing was added to the arrangement. In 2003, Genigraphics closed their 35mm slide E6 photo lab – one of the last high-volume commercial E6 labs in the US – and expanded their large format printing capabilities. Since 2003, Genigraphics has become a major player in the poster session market, providing printing and on-site services to medical and scientific conferences throughout the US and Canada. As of February 2019, over 150,000 medical or scientific ‘ePosters’ are made available through their ResearchPosters.com archive service. === Partnership with Microsoft and development of PowerPoint === As presentations began to be created on personal computers in the late 80’s, Genigraphics sought presentation software partners in Silicon Valley who would be interested in sending files to Genigraphics via dial-up modem to be produced on 35mm slides. In 1987, Michael Beetner, Director of Marketing Planning for Genigraphics, met with Robert Gaskins, head of Microsoft's Graphics Business Unit, who was leading the development of the newly released PowerPoint software. A joint development agreement between Microsoft and Genigraphics was agreed upon and announced at Mac World 1988. According to Erica Robles-Anderson and Patrik Svensson, "It would be hard to overestimate Genigraphics’ influence on PowerPoint. PowerPoint 2.0 was designed for Genigraphics film recorders. It shipped with Genigraphics color palettes, schemes, and the distinctively Genigraphics color-gradient backgrounds. The application contained a ‘Send to Genigraphics’ menu item that wrote the presentation to floppy disk or transmitted the order directly via modem. Within three and a half months PowerPoint orders accounted for ten percent of revenue at Genigraphics service centers. PowerPoint 3.0 was even more intimately dependent upon Genigraphics. The software incorporated a collection of clip art images and symbols that had been produced by hundreds of artists at dozens of service centers across tens of thousands of presentations. Genigraphics artists designed PowerPoint 3.0 colors, templates, and sample presentations. The software even used Genigraphics (rather than Excel) chart style. Bar charts were rendered two-dimensionally with apparent thickness added to make them seemingly recede from the axes. The technique made it easier for viewers to compare bar heights and estimate values from axis ticks and labels. Pie charts were handled analogously. Microsoft paid Genigraphics to produce more than 500 clip art drawings and symbols used in Microsoft programs.” In exchange for Genigraphics development efforts, Microsoft included a “Send to Genigraphics” link in every copy of PowerPoint through the 10.0 version (2000/2001). The arrangement came to an end when Microsoft restructured as a result of anti-trust lawsuits.
What I eat in a day video
"What I eat in a day" videos are a trend on several social media platforms where a person describes all the meals and snacks that they eat during a given day, often as part of a given diet. The videos, shared on platforms including Twitter, TikTok and YouTube, become increasingly popular in 2020, with some of them accumulating millions of views, and they are considered a profitable industry for the people making them. Some have raised concerns that the videos may promote an unrealistic standard for healthy eating and contribute to the development of eating disorders. == Format == These videos often feature a montage of the food that the creator eats over the course of the day, sometimes with the associated calorie count of the foods that they describe. Unlike related mukbang videos, however, in which participants eat large amounts of food, the diets described are often restrictive. However, other videos are labeled as "unhealthy" and depict large portion sizes and higher amounts of processed food. == Popularity == "What I eat in a day" videos have existed for a long time, especially on YouTube, but they have become much more widespread in recent years. This phenomenon is self-reinforcing because when social media users watch or like these videos they are likely to see more of them in the future. Indeed, some of the most successful videos have tens of millions of view each. == Criticism and controversy == Several dieticians and mental health professionals over the impacts that these videos can have, as they can advocate a restrictive style of eating and not "promote body diversity." They have also raised concerns that this trend could contribute to a rise in disordered eating, especially since use of social media is known to increase feelings of negative body image. This trend is particularly prevalent among young adults, which are also the group with the highest vulnerability to eating disorders. More recently, a portion of these videos have begun to challenge diets and depict more realistic ways of eating in order to reduce the potential consequences of the trend.
Cryptographic bill of materials
Cryptographic bill of materials (CBOM—also cryptography bill of materials) is a structured inventory of all cryptographic assets present in a software, firmware, device, or system. It enumerates algorithms (and parameters such as key sizes and modes), cryptographic libraries or modules, digital certificates, keys and related material, and protocols in use, and maps their relationships to the components that implement or invoke them. CBOMs are used to improve security analysis, compliance, and cryptographic agility, and are increasingly referenced in guidance for post‑quantum cryptography (PQC) migration. == Definition and scope == A CBOM inventories cryptographic primitives and materials—such as encryption and signature algorithms (with specific variants and modes), key sizes, cryptographic libraries/modules, digital certificates (e.g., X.509), keys and other related cryptographic material, and security protocols (e.g., TLS, IPsec). It also documents dependencies (for example, an application uses an algorithm provided by a library; a protocol uses several algorithms) and can capture certificate lifecycles, cryptographic module certifications (e.g., FIPS 140‑3), and policy conformance metadata. In common practice, a CBOM may be embedded within an SBOM format (such as CycloneDX) or exported as a separate, linked artifact. === Typical CBOM fields === The exact schema varies by implementation, but common fields are summarized below (see CycloneDX CBOM guide and NIST SP 1800‑38B). == Relation to SBOM == A CBOM is complementary to, but distinct from, a software bill of materials (SBOM). Whereas an SBOM lists software components and their versions, a CBOM focuses specifically on the cryptography present and how it is configured and used. For example, an SBOM might enumerate inclusion of a library such as OpenSSL, while the CBOM would identify which algorithms and parameters that library enables (e.g., RSA‑2048, ECDH P‑256, AES‑GCM) and list relevant keys and certificates. The pairing enables both supply‑chain transparency and cryptographic transparency. == History == The term and practice emerged in the early–mid 2020s alongside software‑supply‑chain transparency and PQC planning. The OWASP CycloneDX standard introduced native CBOM support (v1.6 and later), modeling algorithms, keys, certificates, and protocols as first‑class “cryptographic assets” and providing dependency semantics (uses/implements) between software and cryptography. Open tooling from industry and researchers (e.g., IBM's CBOMkit and related generators/viewers) appeared to automate discovery and representation of cryptographic use in the CycloneDX CBOM schema. == Regulatory and policy context == In the United States, policy has emphasized cryptographic inventories as a prerequisite to PQC migration. The White House's National Security Memorandum 10 (2022) directed a government‑wide transition to quantum‑resistant cryptography; the Office of Management and Budget's M‑23‑02 (November 2022) operationalized this by requiring agencies to submit a prioritized inventory of cryptographic systems (with algorithm and key details) by 4 May 2023 and annually thereafter, and tasked CISA/NSA/NIST to develop automated discovery and inventory strategies. A 2024 Office of the National Cyber Director report reiterated that a “comprehensive cryptographic inventory” is the baseline for PQC planning and must be maintained iteratively with both automated and manual discovery. NIST's NCCoE practice guide (SP 1800‑38B, preliminary draft) provides concrete methods for cryptographic discovery and documentation across enterprises, aligning with CBOM‑style representations. CISA later published a strategy to migrate federal agencies to automated cryptography discovery and inventory tools to support continuous reporting. Separately, NSA, CISA, and NIST issued joint guidance encouraging all organisations to prepare cryptographic inventories and roadmaps for PQC, beyond government environments. == Role in quantum readiness and cryptographic agility == Because large‑scale quantum computing threatens widely used public‑key algorithms (e.g., RSA, ECC), organisations are planning multi‑year transitions to post-quantum cryptography. CBOMs enable that planning by identifying where quantum‑vulnerable algorithms appear, prioritising high‑impact systems, and tracking replacements over time. A machine‑readable CBOM also supports cryptographic agility and incident response: if an algorithm, library, or certificate lifecycle becomes non‑compliant or vulnerable, the CBOM indicates which products and systems are affected and where mitigations must be applied first. == Standards and tooling == CycloneDX (OWASP): Native CBOM modelling (v1.6+) for algorithms, certificates, keys/related material, and protocols, with dependency semantics and examples. The project publishes a CBOM guide and use‑case profiles (e.g., certificate and algorithm inventories). NIST NCCoE SP 1800‑38 series: Practice guides for PQC migration include enterprise cryptographic discovery methods that produce CBOM‑like inventories and integrate multiple discovery tools. Government automation initiatives: Following M‑23‑02, CISA issued a strategy to migrate to automated cryptography discovery and inventory tools to support agency reporting and continuous inventory management. Open‑source and vendor tools: IBM's CBOMkit and related components generate, analyse, and visualise CBOMs; the IBM CBOM specification work was upstreamed into CycloneDX 1.6. === Data model and interchange (example) === CycloneDX provides machine‑readable encodings (JSON/XML) for CBOM content. The example below (subset) shows an application depending on a crypto library that provides the AES‑256‑GCM algorithm, and the application also depends on a leaf X.509 certificate. See the CycloneDX CBOM guide, JSON reference, and the “Implementation details” use‑case for the semantics of `dependsOn` and `provides`. == Relationship to cybersecurity supply chain initiatives == CBOMs complement SBOM‑focused supply‑chain transparency introduced by U.S. Executive Order 14028 and NTIA/NIST SBOM work. SBOMs document software components; CBOMs add detail on embedded cryptography to support risk management, policy compliance (e.g., disallowing deprecated algorithms), and PQC transition planning.
Subliminal channel
In cryptography, subliminal channels are covert channels that can be used to communicate secretly in normal looking communication over an insecure channel. Subliminal channels in digital signature crypto systems were found in 1984 by Gustavus Simmons. Simmons describes how the "Prisoners' Problem" can be solved through parameter substitution in digital signature algorithms. == Examples == An easy example of a narrowband subliminal channel for normal human-language text would be to define that an even word count in a sentence is associated with the bit "0" and an odd word count with the bit "1". The question "Hello, how do you do?" would therefore send the subliminal message "1". The Digital Signature Algorithm has one subliminal broadband and three subliminal narrow-band channels == Improvements == A modification to the Brickell and DeLaurentis signature scheme provides a broadband channel without the necessity to share the authentication key. The Newton channel is not a subliminal channel, but it can be viewed as an enhancement. == Countermeasures == With the help of the zero-knowledge proof and the commitment scheme it is possible to prevent the usage of the subliminal channel. This countermeasure has a 1-bit subliminal channel because for is the problem that a proof can succeed or purposely fail. Another countermeasure can detect, and not prevent, the subliminal usage of the randomness.
Audio inpainting
Audio inpainting (also known as audio interpolation) is an audio restoration task which deals with the reconstruction of missing or corrupted portions of a digital audio signal. Inpainting techniques are employed when parts of the audio have been lost due to various factors such as transmission errors, data corruption or errors during recording. The goal of audio inpainting is to fill in the gaps (i.e., the missing portions) in the audio signal seamlessly, making the reconstructed portions indistinguishable from the original content and avoiding the introduction of audible distortions or alterations. Many techniques have been proposed to solve the audio inpainting problem and this is usually achieved by analyzing the temporal and spectral information surrounding each missing portion of the considered audio signal. Classic methods employ statistical models or digital signal processing algorithms to predict and synthesize the missing or damaged sections. Recent solutions, instead, take advantage of deep learning models, thanks to the growing trend of exploiting data-driven methods in the context of audio restoration. Depending on the extent of the lost information, the inpainting task can be divided in three categories. Short inpainting refers to the reconstruction of few milliseconds (approximately less than 10) of missing signal, that occurs in the case of short distortions such as clicks or clipping. In this case, the goal of the reconstruction is to recover the lost information exactly. In long inpainting instead, with gaps in the order of hundreds of milliseconds or even seconds, this goal becomes unrealistic, since restoration techniques cannot rely on local information. Therefore, besides providing a coherent reconstruction, the algorithms need to generate new information that has to be semantically compatible with the surrounding context (i.e., the audio signal surrounding the gaps). The case of medium duration gaps lays between short and long inpainting. It refers to the reconstruction of tens of millisecond of missing data, a scale where the non-stationary characteristic of audio already becomes important. == Definition == Consider a digital audio signal x {\displaystyle \mathbf {x} } . A corrupted version of x {\displaystyle \mathbf {x} } , which is the audio signal presenting missing gaps to be reconstructed, can be defined as x ~ = m ∘ x {\displaystyle \mathbf {\tilde {x}} =\mathbf {m} \circ \mathbf {x} } , where m {\displaystyle \mathbf {m} } is a binary mask encoding the reliable or missing samples of x {\displaystyle \mathbf {x} } , and ∘ {\displaystyle \circ } represents the element-wise product. Audio inpainting aims at finding x ^ {\displaystyle \mathbf {\hat {x}} } (i.e., the reconstruction), which is an estimation of x {\displaystyle \mathbf {x} } . This is an ill-posed inverse problem, which is characterized by a non-unique set of solutions. For this reason, similarly to the formulation used for the inpainting problem in other domains, the reconstructed audio signal can be found through an optimization problem that is formally expressed as x ^ ∗ = argmin X ^ L ( m ∘ x ^ , x ~ ) + R ( x ^ ) {\displaystyle \mathbf {\hat {x}} ^{}={\underset {\hat {\mathbf {X} }}{\text{argmin}}}~L(\mathbf {m} \circ \mathbf {\hat {x}} ,\mathbf {\tilde {x}} )+R(\mathbf {\hat {x}} )} . In particular, x ^ ∗ {\displaystyle \mathbf {\hat {x}} ^{}} is the optimal reconstructed audio signal and L {\displaystyle L} is a distance measure term that computes the reconstruction accuracy between the corrupted audio signal and the estimated one. For example, this term can be expressed with a mean squared error or similar metrics. Since L {\displaystyle L} is computed only on the reliable frames, there are many solutions that can minimize L ( m ∘ x ^ , x ~ ) {\displaystyle L(\mathbf {m} \circ \mathbf {\hat {x}} ,\mathbf {\tilde {x}} )} . It is thus necessary to add a constraint to the minimization, in order to restrict the results only to the valid solutions. This is expressed through the regularization term R {\displaystyle R} that is computed on the reconstructed audio signal x ^ {\displaystyle \mathbf {\hat {x}} } . This term encodes some kind of a-priori information on the audio data. For example, R {\displaystyle R} can express assumptions on the stationarity of the signal, on the sparsity of its representation or can be learned from data. == Techniques == There exist various techniques to perform audio inpainting. These can vary significantly, influenced by factors such as the specific application requirements, the length of the gaps and the available data. In the literature, these techniques are broadly divided in model-based techniques (sometimes also referred as signal processing techniques) and data-driven techniques. === Model-based techniques === Model-based techniques involve the exploitation of mathematical models or assumptions about the underlying structure of the audio signal. These models can be based on prior knowledge of the audio content or statistical properties observed in the data. By leveraging these models, missing or corrupted portions of the audio signal can be inferred or estimated. An example of a model-based techniques are autoregressive models. These methods interpolate or extrapolate the missing samples based on the neighboring values, by using mathematical functions to approximate the missing data. In particular, in autoregressive models the missing samples are completed through linear prediction. The autoregressive coefficients necessary for this prediction are learned from the surrounding audio data, specifically from the data adjacent to each gap. Some more recent techniques approach audio inpainting by representing audio signals as sparse linear combinations of a limited number of basis functions (as for example in the Short Time Fourier Transform). In this context, the aim is to find the sparse representation of the missing section of the signal that most accurately matches the surrounding, unaffected signal. The aforementioned methods exhibit optimal performance when applied to filling in relatively short gaps, lasting only a few tens of milliseconds, and thus they can be included in the context of short inpainting. However, these signal-processing techniques tend to struggle when dealing with longer gaps. The reason behind this limitation lies in the violation of the stationarity condition, as the signal often undergoes significant changes after the gap, making it substantially different from the signal preceding the gap. As a way to overcome these limitations, some approaches add strong assumptions also about the fundamental structure of the gap itself, exploiting sinusoidal modeling or similarity graphs to perform inpainting of longer missing portions of audio signals. === Data-driven techniques === Data-driven techniques rely on the analysis and exploitation of the available audio data. These techniques often employ deep learning algorithms that learn patterns and relationships directly from the provided data. They involve training models on large datasets of audio examples, allowing them to capture the statistical regularities present in the audio signals. Once trained, these models can be used to generate missing portions of the audio signal based on the learned representations, without being restricted by stationarity assumptions. Data-driven techniques also offer the advantage of adaptability and flexibility, as they can learn from diverse audio datasets and potentially handle complex inpainting scenarios. As of today, such techniques constitute the state-of-the-art of audio inpainting, being able to reconstruct gaps of hundreds of milliseconds or even seconds. These performances are made possible by the use of generative models that have the capability to generate novel content to fill in the missing portions. For example, generative adversarial networks, which are the state-of-the-art of generative models in many areas, rely on two competing neural networks trained simultaneously in a two-player minmax game: the generator produces new data from samples of a random variable, the discriminator attempts to distinguish between generated and real data. During the training, the generator's objective is to fool the discriminator, while the discriminator attempts to learn to better classify real and fake data. In GAN-based inpainting methods the generator acts as a context encoder and produces a plausible completion for the gap only given the available information surrounding it. The discriminator is used to train the generator and tests the consistency of the produced inpainted audio. Recently, also diffusion models have established themselves as the state-of-the-art of generative models in many fields, often beating even GAN-based solutions. For this reason they have also been used to solve the audio inpainting problem, obtaining valid results. These models generate new data instances by inverting the
AS2
AS2 (Applicability Statement 2) is a specification on how to transport structured business-to-business data securely and reliably over the Internet. Security is achieved by using digital certificates and encryption. == Background == AS2 was created in 2002 by the IETF to replace AS1, which they created in the early 1990s. The adoption of AS2 grew rapidly throughout the early 2000s because major players in the retail and fast-moving consumer goods industries championed AS2. Walmart was the first major retailer to require its suppliers to use the AS2 protocol instead of relying on dial-up modems for ordering goods. Amazon, Target, Lowe's, Bed, Bath, & Beyond and thousands of others followed suit. Many other industries use the AS2 protocol, including healthcare, as AS2 meets legal HIPAA requirements. In some cases, AS2 is a way to bypass expensive value-added networks previously used for data interchange. == Technical overview == AS2 is specified in RFC 4130, and is based on HTTP and S/MIME. It was the second AS protocol developed and uses the same signing, encryption and MDN (as defined by RFC3798) conventions used in the original AS1 protocol introduced in the late 1990s by IETF. In other words: Files are encoded as "attachments" in a standardized S/MIME message (an AS2 message). AS2 does not specify the contents of the files. Usually, the file contents are in a standardized format that is separately agreed upon, such as XML or EDIFACT. AS2 messages are always sent using the HTTP or HTTPS protocol (Secure Sockets Layer — also known as SSL — is implied by HTTPS) and usually use the "POST" method (use of "GET" is rare). Messages can be signed, but do not have to be. Messages can be encrypted, but do not have to be. Messages may request a Message Disposition Notification (MDN) back if all went well, but do not have to request such a message. If the original AS2 message requested an MDN: Upon the receipt of the message and its successful decryption or signature validation (as necessary) a "success" MDN will be sent back to the original sender. This MDN is typically signed but never encrypted (unless temporarily encrypted in transit via HTTPS). Upon the receipt and successful verification of the signature on the MDN, the original sender will "know" that the recipient got their message (this provides the "Non-repudiation" element of AS2). If there are any problems receiving or interpreting the original AS2 message, a "failed" MDN may be sent back. However, part of the AS2 protocol states that the client must treat a lack of an MDN as a failure as well, so some AS2 receivers will not return an MDN in this case. Like any other AS file transfer, AS2 file transfers typically require both sides of the exchange to trade X.509 certificates and specific "trading partner" names before any transfers can take place. AS2 trading partner names can usually be any valid phrase. === MDN options === Unlike AS1 or AS3 file transfers, AS2 file transfers offer several "MDN return" options instead of the traditional options of "yes" or "no". Specifically, the choices are: ==== AS2 w/ "Sync" MDNs ==== Return Synchronous MDN via HTTP(S) ("AS2 Sync") - This popular option allows AS2 MDNs to be returned to AS2 message sender clients over the same HTTP connection they used to send the original message. This "MDN while you wait" capability makes "AS2 Sync" transfers the fastest of any type of AS file transfer, but it also keeps this flavor of MDN requests from being used with large files (which may time out in low-bandwidth situations). ==== AS2 w/ "ASync" MDNs ==== Return Asynchronous MDN via HTTP(S) (a.k.a. "AS2 Async") - This popular option allows AS2 MDNs to be returned to the AS2 message sender's server later over a different HTTP connection. This flavor of MDN request is usually used if large files are involved or if your trading partner's AS2 server has poor Internet service. ==== AS2 w/ "Email" MDNs ==== Return (Asynchronous) MDN via Email - This rarely used option allows AS2 MDNs to be returned to AS2 message senders via email rather than HTTP. Otherwise, it is similar to "AS2 Async (HTTP)". ==== AS2 w/ No MDNs ==== Do not return MDN - This option works like it does in any other AS protocol: the receiver of an AS2 message with this option set simply does not try to return an MDN to the AS2 message sender. ==== Filename preservation ==== AS2 filename preservation feature will be used to communicate the filename to the trading partner. The banking industry relies on filenames being communicated between trading partners. AS2 vendors are currently certifying that implementation of filename communication conforms to the standard and is interoperable. There are two profiles for filename preservation being optionally tested under AS2 testing: Filename preservation without MDN responses Filename preservation with an associated MDN response certification Walmart recommends contacting Drummond Group, LLC for more information on EDIINT AS2, or for a list of interoperable-testing AS2 software providers. == Benefits == For many businesses, the use of AS2 and electronic data interchange (EDI) is not a choice so much as it is a requirement of doing business with a large customer or partner. That said, AS2 is a universal protocol that has benefits, from both business and technology vantage points. === Business case === Cut costs by using the web for EDI file transfers, AS2 reduces the cost of transactions from expensive VANs. Extend EDI to more partners; with lower costs and universal web connectivity, AS2 allows organizations to implement EDI with partners worldwide that have little EDI infrastructure. Save time by eliminating the need to manually process orders. Eliminate errors by turning manual processes into automated processes. Universal solution — AS2 is established and tested, so no one has to re-invent the wheel. === Technological advantages === Leverage the web: if an organization can share data securely via the web, they already have much of the infrastructure for AS2. Unlimited EDI data — there are no practical limitations on transaction sizes via the web, and AS2 includes features for managing large transfers. Payload Agnostic — AS2 can be used to transport any type of document. While EDI X12, EDIFACT and XML are common, any mutually agreed-upon format may be transferred.