The cross-entropy (CE) method is a Monte Carlo method for importance sampling and optimization. It is applicable to both combinatorial and continuous problems, with either a static or noisy objective. The method approximates the optimal importance sampling estimator by repeating two phases: Draw a sample from a probability distribution. Minimize the cross-entropy between this distribution and a target distribution to produce a better sample in the next iteration. Reuven Rubinstein developed the method in the context of rare-event simulation, where tiny probabilities must be estimated, for example in network reliability analysis, queueing models, or performance analysis of telecommunication systems. The method has also been applied to the traveling salesman, quadratic assignment, DNA sequence alignment, max-cut and buffer allocation problems. == Estimation via importance sampling == Consider the general problem of estimating the quantity ℓ = E u [ H ( X ) ] = ∫ H ( x ) f ( x ; u ) d x {\displaystyle \ell =\mathbb {E} _{\mathbf {u} }[H(\mathbf {X} )]=\int H(\mathbf {x} )\,f(\mathbf {x} ;\mathbf {u} )\,{\textrm {d}}\mathbf {x} } , where H {\displaystyle H} is some performance function and f ( x ; u ) {\displaystyle f(\mathbf {x} ;\mathbf {u} )} is a member of some parametric family of distributions. Using importance sampling this quantity can be estimated as ℓ ^ = 1 N ∑ i = 1 N H ( X i ) f ( X i ; u ) g ( X i ) {\displaystyle {\hat {\ell }}={\frac {1}{N}}\sum _{i=1}^{N}H(\mathbf {X} _{i}){\frac {f(\mathbf {X} _{i};\mathbf {u} )}{g(\mathbf {X} _{i})}}} , where X 1 , … , X N {\displaystyle \mathbf {X} _{1},\dots ,\mathbf {X} _{N}} is a random sample from g {\displaystyle g\,} . For positive H {\displaystyle H} , the theoretically optimal importance sampling density (PDF) is given by g ∗ ( x ) = H ( x ) f ( x ; u ) / ℓ {\displaystyle g^{}(\mathbf {x} )=H(\mathbf {x} )f(\mathbf {x} ;\mathbf {u} )/\ell } . This, however, depends on the unknown ℓ {\displaystyle \ell } . The CE method aims to approximate the optimal PDF by adaptively selecting members of the parametric family that are closest (in the Kullback–Leibler sense) to the optimal PDF g ∗ {\displaystyle g^{}} . == Generic CE algorithm == Choose initial parameter vector v ( 0 ) {\displaystyle \mathbf {v} ^{(0)}} ; set t = 1. Generate a random sample X 1 , … , X N {\displaystyle \mathbf {X} _{1},\dots ,\mathbf {X} _{N}} from f ( ⋅ ; v ( t − 1 ) ) {\displaystyle f(\cdot ;\mathbf {v} ^{(t-1)})} Solve for v ( t ) {\displaystyle \mathbf {v} ^{(t)}} , where v ( t ) = argmax v 1 N ∑ i = 1 N H ( X i ) f ( X i ; u ) f ( X i ; v ( t − 1 ) ) log f ( X i ; v ) {\displaystyle \mathbf {v} ^{(t)}=\mathop {\textrm {argmax}} _{\mathbf {v} }{\frac {1}{N}}\sum _{i=1}^{N}H(\mathbf {X} _{i}){\frac {f(\mathbf {X} _{i};\mathbf {u} )}{f(\mathbf {X} _{i};\mathbf {v} ^{(t-1)})}}\log f(\mathbf {X} _{i};\mathbf {v} )} If convergence is reached then stop; otherwise, increase t by 1 and reiterate from step 2. In several cases, the solution to step 3 can be found analytically. Situations in which this occurs are When f {\displaystyle f\,} belongs to the natural exponential family When f {\displaystyle f\,} is discrete with finite support When H ( X ) = I { x ∈ A } {\displaystyle H(\mathbf {X} )=\mathrm {I} _{\{\mathbf {x} \in A\}}} and f ( X i ; u ) = f ( X i ; v ( t − 1 ) ) {\displaystyle f(\mathbf {X} _{i};\mathbf {u} )=f(\mathbf {X} _{i};\mathbf {v} ^{(t-1)})} , then v ( t ) {\displaystyle \mathbf {v} ^{(t)}} corresponds to the maximum likelihood estimator based on those X k ∈ A {\displaystyle \mathbf {X} _{k}\in A} . == Continuous optimization—example == The same CE algorithm can be used for optimization, rather than estimation. Suppose the problem is to maximize some function S {\displaystyle S} , for example, S ( x ) = e − ( x − 2 ) 2 + 0.8 e − ( x + 2 ) 2 {\displaystyle S(x)={\textrm {e}}^{-(x-2)^{2}}+0.8\,{\textrm {e}}^{-(x+2)^{2}}} . To apply CE, one considers first the associated stochastic problem of estimating P θ ( S ( X ) ≥ γ ) {\displaystyle \mathbb {P} _{\boldsymbol {\theta }}(S(X)\geq \gamma )} for a given level γ {\displaystyle \gamma \,} , and parametric family { f ( ⋅ ; θ ) } {\displaystyle \left\{f(\cdot ;{\boldsymbol {\theta }})\right\}} , for example the 1-dimensional Gaussian distribution, parameterized by its mean μ t {\displaystyle \mu _{t}\,} and variance σ t 2 {\displaystyle \sigma _{t}^{2}} (so θ = ( μ , σ 2 ) {\displaystyle {\boldsymbol {\theta }}=(\mu ,\sigma ^{2})} here). Hence, for a given γ {\displaystyle \gamma \,} , the goal is to find θ {\displaystyle {\boldsymbol {\theta }}} so that D K L ( I { S ( x ) ≥ γ } ‖ f θ ) {\displaystyle D_{\mathrm {KL} }({\textrm {I}}_{\{S(x)\geq \gamma \}}\|f_{\boldsymbol {\theta }})} is minimized. This is done by solving the sample version (stochastic counterpart) of the KL divergence minimization problem, as in step 3 above. It turns out that parameters that minimize the stochastic counterpart for this choice of target distribution and parametric family are the sample mean and sample variance corresponding to the elite samples, which are those samples that have objective function value ≥ γ {\displaystyle \geq \gamma } . The worst of the elite samples is then used as the level parameter for the next iteration. This yields the following randomized algorithm that happens to coincide with the so-called Estimation of Multivariate Normal Algorithm (EMNA), an estimation of distribution algorithm. === Pseudocode === // Initialize parameters μ := −6 σ2 := 100 t := 0 maxits := 100 N := 100 Ne := 10 // While maxits not exceeded and not converged while t < maxits and σ2 > ε do // Obtain N samples from current sampling distribution X := SampleGaussian(μ, σ2, N) // Evaluate objective function at sampled points S := exp(−(X − 2) ^ 2) + 0.8 exp(−(X + 2) ^ 2) // Sort X by objective function values in descending order X := sort(X, S) // Update parameters of sampling distribution via elite samples μ := mean(X(1:Ne)) σ2 := var(X(1:Ne)) t := t + 1 // Return mean of final sampling distribution as solution return μ == Related methods == Simulated annealing Genetic algorithms Harmony search Estimation of distribution algorithm Tabu search Natural Evolution Strategy Ant colony optimization algorithms
Sports Card Investor
Sports Card Investor is an American sports collectibles media platform and mobile application founded by Geoff Wilson. The platform provides market data, analysis, and editorial content focused on sports trading cards and related collectibles. It operates a website, mobile app, and digital media channels covering developments in the sports card industry. The company posted its first YouTube video in July 2019, shortly before a period of rapid growth in sports card collecting in the early 2020s, which was marked by increased trading volumes and mainstream media attention. == History == Sports Card Investor was founded by Geoff Wilson, an entrepreneur and collector who began publishing sports card–related content online before launching the platform's dedicated app and subscription tools. In February 2020, the company launched Market Movers, the first website and app to chart sports card prices and track card collections. The platform expanded its media presence through partnerships and distribution agreements. In 2023, Yahoo Sports announced a new collectibles coverage initiative that included additional content from Sports Card Investor. In February 2024, the Sports Card Investor studio relocated to CardsHQ in Atlanta, Georgia, and visitors to the facility can watch Sports Card Investor videos being filmed. == Platform and content == The Sports Card Investor app provides users with pricing data, portfolio-tracking tools, and market-trend analysis for trading cards. The company also produces video and editorial content discussing market developments, grading trends, and major card releases. Coverage in industry publications has referenced Sports Card Investor in discussions about shifts in sports card licensing rights and hobby market reactions. == Industry context == The growth of Sports Card Investor coincided with a broader resurgence in trading card markets, including record sales and expanded retail presence. Mainstream outlets have cited the company and its founder in reporting on collectibles investing trends, grading practices, and market volatility. The Sports Card Investor app has attracted over 37,000 reviews on the Apple App Store, reflecting its strong user engagement within the sports card community.
BFR algorithm
The BFR algorithm, named after its inventors Bradley, Fayyad and Reina, is a variant of k-means algorithm that is designed to cluster data in a high-dimensional Euclidean space. It makes a very strong assumption about the shape of clusters: they must be normally distributed about a centroid. The mean and standard deviation for a cluster may differ for different dimensions, but the dimensions must be independent. In other words, the data must take the shape of axis-aligned ellipses.
Co-occurrence
In linguistics, co-occurrence or cooccurrence (in older texts often shown with diacritic as coöccurrence) is an above-chance frequency of ordered occurrence of two adjacent terms in a text corpus. Co-occurrence in this linguistic sense can be interpreted as an indicator of semantic proximity or an idiomatic expression. Corpus linguistics and its statistical analyses can reveal (regularity of) patterns of co-occurrences within a language and enable the working out of typical collocations for its lexical items. A co-occurrence restriction is identified when linguistic elements never occur together. Analysis of these restrictions can lead to discoveries about the structure and development of a language. Co-occurrence can be seen an extension of word counting in higher dimensions. Co-occurrence can be quantitatively described using measures like a massive correlation or mutual information. Co-occurrence information and knowledge of co-occurring words may be relevant in analysis of language for the purposes of large language models, part of the emerging field of artificial intelligence, and helpful in word games such as scrabble.
Bruno Zamborlin
Bruno Zamborlin (born 1983 in Vicenza) is an AI researcher, entrepreneur and artist based in London, working in the field of human-computer interaction. His work focuses on converting physical objects into touch-sensitive, interactive surfaces using vibration sensors and artificial intelligence. In 2013, he founded Mogees Limited a start-up to transform everyday objects into musical instruments and games using a vibration sensor and a mobile phone. With HyperSurfaces, he converts physical surfaces of any material, shape and form into data-enabled-interactive surfaces using a vibration sensor and a coin-sized chipset. As an artist, he has created art installations around the world, with his most recent work comprising a unique series of "sound furnitures" that was showcased at the Italian Pavilion of the Venice Biennale 2023. He regularly performed with UK-based electronic music duo Plaid (Warp Records). He is also honorary visiting research fellow at Goldsmiths, University of London. == Early life and education == From 2008-2011, Zamborlin worked at the IRCAM (Institute for Research and Coordination Acoustic Musical) – Centre Pompidou as a member of the Sound Music Movement Interaction team. Under the supervision of Frederic Bevilacqua, he started experimenting with the use of artificial intelligence and human movements, and contributed to the creation of Gesture Follower, a software used to analyse body movements of performers and dancers through motion sensors in order to control sound and visual media in real-time, slowing down or speeding up their reproduction based on the speed the gestures are performed. He has lived in London since 2011, where he developed a joint PhD between Goldsmiths, University of London and IRCAM - Centre Pompidou/Pierre and Marie Curie University Paris in AI, focussing on the concept of Interactive Machine Learning applied to digital musical instruments and performing arts. == Career == Zamborlin founded Mogees Limited in 2013 in London, with IRCAM being amongst the early partners. Mogees transform physical objects into musical instruments and games using a vibration sensor and a series of apps for smartphones and desktop. After a campaign on Kickstarter in 2014, Mogees was used both by common users and artists such as Rodrigo y Gabriela, Jean-Michel Jarre and Plaid. The algorithms implemented in these apps employ a special version of physical modelling sound synthesis, where the vibration produced by users when interacting with the physical object are used as exciter for a digital resonator which runs in the app. The result is a hybrid, half acoustic and half digital sound which is a function of both software and acoustic properties of the physical object the users decide to play. In 2017, Zamborlin founded HyperSurfaces together with computational artist Parag K Mital. to merge "the physical and the digital worlds". HyperSurfaces technology converts any surface made of any material, shape and size into data-enabled interactive objects, employing a vibration sensor and proprietary AI algorithms running on a coin-sized chipset. The vibrations generated by people's interactions on the surface are converted into an electric signal by a piezoelectric sensor and analysed in realtime by AI algorithms that run on the chipset. Anytime the AI recognises in the vibration signal one of the events that have been predefined by the user beforehand, a corresponding notification message is generated in realtime and sent to some application. The technology can be applied to anything ranging from button-less human-computer interaction applications for automotive and smart home to the Internet of things. Because the AI algorithms employed by HyperSurfaces run locally on a chipset, without the need to access cloud-based services, they are considered to be part of the field of edge computing. Also, because the AI can be trained beforehand to recognise the events its users are interested in, HyperSurfaces algorithms belong to the field of supervised machine learning. == Selected awards == IRISA Prix Jeune Chercheur, 13 October 2012 NeMoDe, New Economic Models in the Digital Economy, 25 October 2012 == Patents and academic publications == United States pending US10817798B2, Bruno Zamborlin & Carmine Emanuele Cella, "Method to recognize a gesture and corresponding device", published 27 April 2016, assigned to Mogees Limited GB Pending WO/2019/086862, Bruno Zamborlin; Conor Barry & Alessandro Saccoia et al., "A user interface for vehicles", published 9 May 2019, assigned to Mogees Limited GB Pending WO/2019/086863, Bruno Zamborlin; Conor Barry & Alessandro Saccoia et al., "Trigger for game events", published 9 May 2019, assigned to Mogees Limited Bevilacqua, Frédéric; Zamborlin, Bruno; Sypniewski, Anthony; Schnell, Norbert; Guédy, Fabrice; Rasamimanana, Nicolas (2010). "Continuous Realtime Gesture Following and Recognition". Gesture in Embodied Communication and Human-Computer Interaction. Lecture Notes in Computer Science. Vol. 5934. pp. 73–84. doi:10.1007/978-3-642-12553-9_7. ISBN 978-3-642-12552-2. S2CID 16251822. Retrieved 17 January 2021. Rasamimanana, Nicolas; Bevilacqua, Frédéric; Schnell, Norbert; Guédy, Fabrice; Flety, Emmanuel; Maestracci, Come; Zamborlin, Bruno (January 2010). "Modular musical objects towards embodied control of digital music". Proceedings of the fifth international conference on Tangible, embedded, and embodied interaction. Tei '11. pp. 9–12. doi:10.1145/1935701.1935704. ISBN 9781450304788. S2CID 10782645. Retrieved 17 January 2021. Bevilacqua, Frédéric; Schnell, Norbert; Rasamimanana, Nicolas; Zamborlin, Bruno; Guedy, Fabrice (2011). "Online Gesture Analysis and Control of Audio Processing". Musical Robots and Interactive Multimodal Systems. Springer Tracts in Advanced Robotics. Vol. 74. pp. 127–142. doi:10.1007/978-3-642-22291-7_8. ISBN 978-3-642-22290-0. Retrieved 17 January 2021. Zamborlin, Bruno; Bevilacqua, Frédéric; Gillies, Marco; D'Inverno, Mark (15 January 2014). "Fluid gesture interaction design: Applications of continuous recognition for the design of modern gestural interfaces". ACM Transactions on Interactive Intelligent Systems. 3 (4): 22:1–22:30. doi:10.1145/2543921. S2CID 7887245. Retrieved 17 January 2021. Leslie, Grace; Zamborlin, Bruno; Schnell, Norbert; Jodlowski, Pierre (15 June 2010). "A Collaborative, Interactive Sound Installation". Proceedings of the International Computer Music Conference. Retrieved 17 January 2021. Kimura, Mari; Rasamimanana, Nicolas; Bevilacqua, Frédéric; Zamborlin, Bruno; Schnell, Bruno; Flety, Emmanuel (2012). "Extracting Human Expression For Interactive Composition with the Augmented Violin". International Conference on New Interfaces for Musical Expression. Retrieved 17 January 2021. Ferretti, Stefano; Roccetti, Marco; Zamborlin, Bruno (13 January 2009). "On SPAWC: Discussion on a Musical Signal Parser and Well-Formed Composer". 2009 6th IEEE Consumer Communications and Networking Conference. pp. 1–5. doi:10.1109/CCNC.2009.4784966. ISBN 978-1-4244-2308-8. S2CID 14213587. Zamborlin, Bruno; Partesana, Giorgio; Liuni, Marco (15 May 2011). "(LAND)MOVES". Conference on New Interfaces for Musical Expression, NIME: 537–538. Retrieved 17 January 2021.
Dropbox Paper
Dropbox Paper, or simply Paper, is a collaborative document-editing service developed by Dropbox. Originating from the company's acquisition of document collaboration company Hackpad in April 2014, Dropbox Paper was officially announced in October 2015, and launched in January 2017. It offers a web application, as well as mobile apps for Android and iOS. Dropbox Paper was described in the official announcement post as "a flexible workspace that brings people and ideas together. With Paper, teams can create, review, revise, manage, and organize — all in shared documents". Reception of Dropbox Paper has been mixed. Critics praised collaboration functionality, including content available immediately, the ability to mention specific collaborators, assign tasks, write comments, as well as editing attribution, and revision history. It received particular praise for its support for rich media from a variety of sources, with one reviewer noting that the Paper's support for rich media exceeds the capabilities of most of its competitors. However, it was criticized for a lack of formatting options and editing features. While the user interface was liked for being minimal, reviewers cited the lack of a fixed formatting bar and missing features present in competitors' products as making Dropbox Paper seem like a "light" tool. == History == Dropbox acquired document collaboration company Hackpad in April 2014. A year later, Dropbox launched a Dropbox Notes note-taking product in beta testing phase. Dropbox Paper was officially announced on October 15, 2015, followed by an open beta and release of mobile Android and iOS apps in August 2016. Dropbox Paper was officially released on January 30, 2017. == Reception == In a comparison between Dropbox Paper and Evernote, PC World's Michael Ansaldo wrote that "With its emphasis on document creation, you might expect formatting to be front and center in Dropbox Paper. That's not the case." Ansaldo noted the lack of a "fixed formatting toolbar as you'd find in Evernote or a word processor like Google Docs or Microsoft Word. Instead, the text editor appears as a floating ribbon only when you highlight selected text." The only formatting options available for emphasis were bolding, strikethrough, bulleted and numbered lists, and H1 and H2 tags. Users can also add links, convert text to checklists, and add comments. Ansaldo wrote that "Both Evernote and Dropbox Paper make it easy to add images to a document", but also noted that "Dropbox Paper doesn't support any image editing". Paper supports rich media, and users can "add rich content to your document just by pasting a link to the file. In addition to Dropbox, Paper supports media from a variety of popular services including YouTube, Spotify, Vimeo, SoundCloud, Facebook, and Google's productivity suite. Once the file appears, you can delete the link for a cleaner display." To start working with other people, Paper "allows you to invite people via email from within a document", with sharing options for who can view the link (anyone with the link or just the invited person), and action permissions (edit or only comment). Regarding collaboration, Ansaldo wrote that "Creative collaboration is Paper’s marquee feature, and it provides a variety of ways to work effectively with others in real time". Users can "make any content immediately visible and accessible to a specific collaborator with "@mentions"", and "You can also use @mentions to create and assign task lists within a document." Paper also "boasts essential collaboration tools including comments, editing attribution, and revision history." Writing for TechRadar, John Brandon wrote that Dropbox Paper "might be a 'light' tool for now without the extensive templates of Microsoft Office or the integration with other apps in the Zoho suite, but it does work well with the Dropbox storage service that's so popular with office workers these days." Kyle Wiggers of Digital Trends wrote that Paper is "all about minimizing distractions. Its interface is quite literally a big, blank canvas on which you tap out your agenda. You can organize notes by title and create to-do lists, but even basic formatting tools are obscured from view", noting Paper's "floating box above words and phrases highlighted by your cursor". Wiggers stated that "Paper is not a to-do organizer", but that it's "well suited to the purpose thanks to a bevy of labor-saving conveniences", highlighting that Paper "supports more media than most of its to-do and note-taking counterparts". He praised the collaboration tools, writing that they "are as extensive as you'd hope, and then some", citing its invitation system with permission controls, lists of changes and revision history, comment and chat support, and "perhaps best of all", the ability to assign tasks with a "@" mention. Business Insider's Alex Heath praised that "Paper's interface is spotless and friendly to write in. You don't feel overwhelmed with formatting options", but criticized the available features, writing that "Google Docs is much more full-featured in the formatting department, so Paper has some catching up to do if it wants to be on par with the competition". Writing for The Verge, Casey Newton praised Paper's handling of rich media, complimenting it for being "great", and added that "I imagine that creative types who work on teams will appreciate having rich media embedded in the documents they're working on rather than in a series of infinite tabs".
Best AI Marketing Tools in 2026
Trying to pick the best AI marketing tool? An AI marketing tool is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right AI marketing tool slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.