Lumpers and splitters are opposing factions in any academic discipline that has to place individual examples into rigorously defined categories. The lumper–splitter problem occurs when there is the desire to create classifications and assign examples to them, for example, schools of literature, biological taxa, and so on. A "lumper" is a person who assigns examples broadly, judging that differences are not as important as signature similarities. A "splitter" makes precise definitions, and creates new categories to classify samples that differ in key ways. == Origin of the terms == The earliest known use of these terms was thought to be by Charles Darwin, in a letter to Joseph Dalton Hooker in 1857: "It is good to have hair-splitters & lumpers". But according to research done by the deputy director at NCSE, Glenn Branch, the credit is due to naturalist Edward Newman who wrote in 1845, "The time has arrived for discarding imaginary species, and the duty of doing this is as imperative as the admission of new ones when such are really discovered. The talents described under the respective names of 'hair-splitting' and 'lumping' are unquestionably yielding their power to the mightier power of Truth." They were then introduced more widely by George G. Simpson in his 1945 work The Principles of Classification and a Classification of Mammals. As he put it: splitters make very small units – their critics say that if they can tell two animals apart, they place them in different genera ... and if they cannot tell them apart, they place them in different species. ... Lumpers make large units – their critics say that if a carnivore is neither a dog nor a bear, they call it a cat. A later use can be found in the title of a 1969 paper "On lumpers and splitters ..." by the medical geneticist Victor McKusick. Reference to lumpers and splitters in the humanities appeared in a debate in 1975 between J. H. Hexter and Christopher Hill, in the Times Literary Supplement. It followed from Hexter's detailed review of Hill's book Change and Continuity in Seventeenth Century England, in which Hill developed Max Weber's argument that the rise of capitalism was facilitated by Calvinist Puritanism. Hexter objected to Hill's "mining" of sources to find evidence that supported his theories. Hexter argued that Hill plucked quotations from sources in a way that distorted their meaning. Hexter explained this as a mental habit that he called "lumping". According to him, "lumpers" rejected differences and chose to emphasise similarities. Any evidence that did not fit their arguments was ignored as aberrant. Splitters, by contrast, emphasised differences, and resisted simple schemes. While lumpers consistently tried to create coherent patterns, splitters preferred incoherent complexity. == Usage in various fields == === Biology === The categorisation and naming of a particular species should be regarded as a hypothesis about the evolutionary relationships and distinguishability of that group of organisms. As further information comes to hand, the hypothesis may be confirmed or refuted. Sometimes, especially in the past when communication was more difficult, taxonomists working in isolation have given two distinct names to individual organisms later identified as the same species. When two named species are agreed to be of the same species, the older species name is almost always retained dropping the newer species name honouring a convention known as "priority of nomenclature". This form of lumping is technically called synonymisation. Dividing a taxon into multiple, often new, taxa is called splitting. Taxonomists are often referred to as "lumpers" or "splitters" by their colleagues, depending on their personal approach to recognizing differences or commonalities between organisms. For example, the number of genera used in Pteridophyte Phylogeny Group I (PPG I) has proved controversial. PPG I uses 18 lycophyte and 319 fern genera. The earlier system put forward by Smith et al. (2006) had suggested a range of 274 to 312 genera for ferns alone. By contrast, the system of Christenhusz & Chase (2014) used 5 lycophyte and about 212 fern genera. The number of fern genera was further reduced to 207 in a subsequent publication. Defending PPG I, Schuettpelz et al. (2018) argue that the larger number of genera is a result of "the gradual accumulation of new collections and new data" and hence "a greater appreciation of fern diversity and ... an improved ability to distinguish taxa". They also argue that the number of species per genus in the PPG I system is already higher than in other groups of organisms (about 33 species per genus for ferns as opposed to about 22 species per genus for angiosperms) and that reducing the number of genera as Christenhusz and Chase propose yields the excessive number of about 50 species per genus for ferns. In response, Christenhusz and Chase (2018) argue that the excessive splitting of genera destabilises the usage of names and will lead to greater instability in future, and that the highly split genera have few if any characters that can be used to recognise them, making identification difficult, even to generic level. They further argue that comparing numbers of species per genus in different groups is "fundamentally meaningless". === History === In history, lumpers are those who tend to create broad definitions that cover large periods of time and many disciplines, whereas splitters want to assign names to tight groups of inter-relationships. Lumping tends to create a more and more unwieldy definition, with members having less and less mutually in common. This can lead to definitions which are little more than conventionalities, or groups which join fundamentally different examples. Splitting often leads to "distinctions without difference", ornate and fussy categories, and failure to see underlying similarities. For example, in the arts, "Romantic" can refer specifically to a period of German poetry roughly from 1780 to 1810, but would exclude the later work of Goethe, among other writers. In music it can mean every composer from Hummel through Rachmaninoff, plus many that came after. === Software modelling === Software engineering often proceeds by building models (sometimes known as model-driven architecture). A lumper is keen to generalise, and produces models with a small number of broadly defined objects. A splitter is reluctant to generalise, and produces models with a large number of narrowly defined objects. Conversion between the two styles is not necessarily symmetrical. For example, if error messages in two narrowly defined classes behave in the same way, the classes can be easily combined. But if some messages in a broad class behave differently, every object in the class must be examined before the class can be split. This illustrates the principle that "splits can be lumped more easily than lumps can be split". === Language classification === There is no agreement among historical linguists about what amount of evidence is needed for two languages to be safely classified in the same language family. For this reason, many proposed language families have had lumper–splitter controversies, including Altaic, Pama–Nyungan, Nilo-Saharan, and most of the larger families of the Americas. At a completely different level, the splitting of a mutually intelligible dialect continuum into different languages, or lumping them into one, is also an issue that continually comes up, though the consensus in contemporary linguistics is that there is no completely objective way to settle the question. Splitters regard the comparative method (meaning not comparison in general, but only reconstruction of a common ancestor or protolanguage) as the only valid proof of kinship, and consider genetic relatedness to be the question of interest. American linguists of recent decades tend to be splitters. Lumpers are more willing to admit techniques like mass lexical comparison or lexicostatistics, and mass typological comparison, and to tolerate the uncertainty of whether relationships found by these methods are the result of linguistic divergence (descent from common ancestor) or language convergence (borrowing). Much long-range comparison work has been from Russian linguists belonging to the Moscow School of Comparative Linguistics, most notably Vladislav Illich-Svitych and Sergei Starostin. In the United States, Greenberg and Ruhlen's work has been met with little acceptance from linguists. Earlier American linguists like Morris Swadesh and Edward Sapir also pursued large-scale classifications like Sapir's 1929 scheme for the Americas, accompanied by controversy similar to that today. === Religious studies === Paul F. Bradshaw suggests that the same principles of lumping and splitting apply to the study of early Christian liturgy. Lumpers, who tend to predominate in this field, try to find a single line of successive texts from the apostolic age to the
Deep Zoom
Deep Zoom is a technology developed by Microsoft for efficiently transmitting and viewing images. It allows users to pan around and zoom in on a large, high resolution image or a large collection of images. It reduces the time required for initial load by downloading only the region being viewed or only at the resolution it is displayed at. Subsequent regions are downloaded as the user pans to (or zooms into) them; animations are used to hide any jerkiness in the transition. The libraries are also available in other platforms including Java and Flash. == History == The Deep Zoom file format is very similar to the Google Maps image format where images are broken into tiles and then displayed as required. The tiling typically follows a quadtree pattern of increasing resolution of image (in other words twice the zoom and twice the resolution). The main difference is that with Google Maps the actual details on the image change from one zoom level to another, while with Deep Zoom the same image is displayed at each zoom level. Seadragon Software, formerly Sand Codex, first created the Seadragon technology and its implementation of what is now called Deep Zoom. This technology was then absorbed into the Microsoft Live Labs when Seadragon Software was acquired. Engineers from Seadragon now work with Microsoft to integrate their work into technology such as Silverlight and Photosynth. == Deep Zoom examples == The most famous implementation of Deep Zoom was probably the first: the memorabilia collection at the Hard Rock website. Conceived and designed by Duncan/Channon and built by Vertigo, it was demonstrated for the first time in March 2008 at the Microsoft MIX convention in Las Vegas. In 2010, Microsoft Live Labs partnered with the University of California, Berkeley to create ChronoZoom, a DeepZoom-powered time visualization tool that pushed the limits of DeepZoom, since it required zooming from the scale of 13 billion years down to a single day. The project has since graduated to development under Microsoft Research. Another example is the Deep Earth project. It is described by its creators as "a community project focused on creating a rich interactive mapping control using Silverlight2 Deep Zoom. Concentrating on Microsoft Virtual Earth imagery and data the project offers team members the opportunity to learn and share while creating something cool and useful." A paintings collection project http://galleryzoom.co.uk/ shows 1000 high resolution/sensor images individually indexed. (Using Deep Zoom Composer). Blaise Aguera y Arcas gave a demonstration of Seadragon and Photosynth at the 2007 TED conference. In November 2009, 352 Media Group, a Silverlight developer in the Microsoft Silverlight Partner Program, created an example of Deep Zoom using Microsoft Silverlight version 3. It is online at 352 Media Group's Web site. The Winston Churchill Deep Zoom Archived 2010-07-04 at the Wayback Machine mosaic, created by Silverlight developers Shoothill, features as both an online interactive deep zoom and a standalone deep zoom which forms part of the Churchill exhibit in the Churchill War Rooms in Whitehall. In 2010, Shoothill built the Sumatran Tiger Deep Zoom - the largest seen to date - for worldwide conservation charity Fauna and Flora International, featuring thousands of images of endangered species. An early example of Deep Zoom-like technology was implemented at The Department of Maori Affairs in New Zealand in 1997. The technology was used to display Maori land ownership. == Deep Zoom images == The file format used by Deep Zoom (as well as Photosynth and Seadragon Ajax) is XML based. Users can specify a single large image (dzi) or a collection of images (dzc). It also allows for "Sparse Images"; where some parts of the image have greater resolution than others, an example of which can be found on the Seadragon Ajax home page; The bike image displayed is a sparse image. Though used in the proprietary Deep Zoom, the dzi format is open and able to be used by anyone. === Deep Zoom image (dzi) === A DZI has two parts: a DZI file (with either a .dzi or .xml extension) and a subdirectory of image folders. Each folder in the image subdirectory is labeled with its level of resolution. Higher numbers correspond to a higher resolution level; inside each folder are the image tiles corresponding to that level of resolution, numbered consecutively in columns from top left to bottom right. === Deep Zoom collection (dzc) === A DZC is a collection of some number of DZIs linked and referenced by a DZC file (with either a .dzc or .xml extension). At a high level, a collection is a number of image thumbnails whose location is kept track of by the .dzc/.xml file, when zooming into an image, it accesses greater resolutions tiles. A DZC's structure is similar to that of a DZI; the .dzc/.xml file defines the collection and the subdirectory of folders maps to the DZI file structure, each with their set of .dzi/.xml and image tiles. The DZC is used in Microsoft's Pivot, but not in SeaDragon per se. === Sparse Images === Sparse images are a sub-classification of the DZI file type. A sparse image is normally a number of separate photographs with varying resolution levels that have been placed in a single DZI instead of a DZC. Sparse images have no different file structure than that of a DZI and differ only in that there is not a single "highest resolution" level for the entire DZI. == Software that uses Deep Zoom == Image Composite Editor - image stitching tool created by Microsoft Research Deep Zoom Composer - collage maker and simple panorama tool created by Microsoft. Images' resolution is maintained when exporting for web use (via Silverlight Deep Zoom or JavaScript using a third-party template). No longer available for download from Microsoft though it can be found on various other sources such as Internet Archive. == iPhone OS development == Microsoft Live Labs has created an application for the App Store called Seadragon Mobile. It is run over the internet and includes Deep Zoom on the following categories; art, history, maps, photos, Photosynth which anybody can upload to, space and technology & web.
AI Virtual Assistants Reviews: What Actually Works in 2026
Curious about the best AI virtual assistant? An AI virtual assistant is software that uses machine learning to help you get more done — it combines speed, accuracy, and an interface that just works. Hands-on testing shows real-world results vary, so a short free trial is the smartest way to decide. Whether you are a beginner or a pro, the right AI virtual assistant slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.
Lorien Pratt
Lorien Pratt is an American computer scientist known for contributions to transfer learning and for her work in promoting and developing the concept of decision intelligence. She is chief scientist and founder of Quantellia. Since 1988, she has conducted research on the use of machine learning as an academic, professor, industry analyst, and practicing data scientist. Pratt received her AB degree in computer science from Dartmouth College and her master's and doctorate degrees in computer science from Rutgers University. == Learning to Learn == She is best known for her book "Learning to Learn," co-edited with Sebastian Thrun, which provided an overview on how to use machine learning to better understand bias and generalization of discrete subjects. This approach, still largely theoretical when the book was published in 1998, is also called metalearning and is now a foundational underpinning of machine learning algorithms such as GPT-3 and DALL-E. == Research == === Transfer learning === Pratt's research includes early work in transfer learning where she developed the discriminability-based transfer (DBT) algorithm in 1993 during her tenure as a professor of computer science at Colorado School of Mines. This paper is considered one of the earliest academic works referring to the use of transfer in machine learning and has been cited over 400 times as foundational research for deep neural networks. === Decision intelligence === Since then, Pratt's research has continued to explore the relationships between machine learning and human cognition with the concept of decision intelligence, an emerging field of machine learning guided analytics designed to support human decision. Pratt introduced this concept in 2008, and this term has since been used by a number of vendors providing machine learning-guided analytics including Diwo, Peak AI, Sisu, and Tellius as the technologies used to support machine learning at scale have become easier to deploy, manage, and embed into software platforms. Pratt's work is cited as a core starting point for defining modern aspects of decision intelligence. Pratt's work at Quantellia since 2020 has focused on the use of decision intelligence to improve COVID-19-based outcomes.
Internettolken
Internettolken (or InternetPreter) is a web-based machine translating tool. As the first Swedish online translating service, it was started in 2002 and included the English and Swedish languages. Today, there are 14 languages with more than 120 possible combinations. The service is free up to 150 words per day, and as a 2,000-word free testing account. It is available both on its website, and as a gadget on iGoogle. The interface is either English or Swedish. Being a dictionary-based tool, with its own translation software, it can sometimes offer a more accurate translation than Google Translate and others, although the grammar will be incorrect. == Languages currently available ==
Reasoning model
A reasoning model, also known as a reasoning language model (RLM) or large reasoning model (LRM), is a type of large language model (LLM) that has been specifically trained to solve complex tasks requiring multiple steps of logical reasoning. These models demonstrate superior performance on logic, mathematics, and programming tasks compared to standard LLMs. They possess the ability to revisit and revise earlier reasoning steps and utilize additional computation during inference as a method to scale performance, complementing traditional scaling approaches based on training data size, model parameters, and training compute. == Overview == Unlike traditional language models that generate responses immediately, reasoning models allocate additional compute, or thinking, time before producing an answer to solve multi-step problems. OpenAI introduced this terminology in September 2024 when it released the o1 series, describing the models as designed to "spend more time thinking" before responding. The company framed o1 as a reset in model naming that targets complex tasks in science, coding, and mathematics, and it contrasted o1's performance with GPT-4o on benchmarks such as AIME and Codeforces. Independent reporting the same week summarized the launch and highlighted OpenAI's claim that o1 automates chain-of-thought style reasoning to achieve large gains on difficult exams. In operation, reasoning models generate internal chains of intermediate steps, then select and refine a final answer. OpenAI reported that o1's accuracy improves as the model is given more reinforcement learning during training and more test-time compute at inference. The company initially chose to hide raw chains and instead return a model-written summary, stating that it "decided not to show" the underlying thoughts so researchers could monitor them without exposing unaligned content to end users. Commercial deployments document separate "reasoning tokens" that meter hidden thinking and a control for "reasoning effort" that tunes how much compute the model uses. These features make the models slower than ordinary chat systems while enabling stronger performance on difficult problems. == History == The research trajectory toward reasoning models combined advances in supervision, prompting, and search-style inference. Early alignment work on reinforcement learning from human feedback showed that models can be fine-tuned to follow instructions with "human feedback" and preference-based rewards. In 2022, Google Research scientists Jason Wei and Denny Zhou showed that chain-of-thought prompting "significantly improves the ability" of large models on complex reasoning tasks. Input → Step 1 → Step 2 → ⋯ → Step n ⏟ Reasoning chain → Answer {\displaystyle {\text{Input}}\rightarrow \underbrace {{\text{Step}}_{1}\rightarrow {\text{Step}}_{2}\rightarrow \cdots \rightarrow {\text{Step}}_{n}} _{\text{Reasoning chain}}\rightarrow {\text{Answer}}} A companion result demonstrated that the simple instruction "Let's think step by step" can elicit zero-shot reasoning. Follow-up work introduced self-consistency decoding, which "boosts the performance" of chain-of-thought by sampling diverse solution paths and choosing the consensus, and tool-augmented methods such as ReAct, a portmanteau of Reason and Act, that prompt models to "generate both reasoning traces" and actions. Research then generalized chain-of-thought into search over multiple candidate plans. The Tree-of-Thoughts framework from Princeton computer scientist Shunyu Yao proposes that models "perform deliberate decision making" by exploring and backtracking over a tree of intermediate thoughts. OpenAI's reported breakthrough focused on supervising reasoning processes rather than only outcomes, with Lightman et al.'s "Let's Verify Step by Step" reporting that rewarding each correct step "significantly outperforms outcome supervision" on challenging math problems and improves interpretability by aligning the chain-of-thought with human judgment. OpenAI's o1 announcement ties these strands together with a large-scale reinforcement learning algorithm that trains the model to refine its own chain of thought, and it reports that accuracy rises with more training compute and more time spent thinking at inference. Together, these developments define the core of reasoning models. They use supervision signals that evaluate the quality of intermediate steps, they exploit inference-time exploration such as consensus or tree search, and they expose controls for how much internal thinking compute to allocate. OpenAI's o1 family made this approach available at scale in September 2024 and popularized the label "reasoning model" for LLMs that deliberately think before they answer. The development of reasoning models illustrates Richard S. Sutton's "bitter lesson" that scaling compute typically outperforms methods based on human-designed insights. This principle was demonstrated by researchers at the Generative AI Research Lab (GAIR), who initially attempted to replicate o1's capabilities using sophisticated methods including tree search and reinforcement learning in late 2024. Their findings, published in the "o1 Replication Journey" series, revealed that knowledge distillation, a comparatively straightforward technique that trains a smaller model to mimic o1's outputs, produced unexpectedly strong performance. This outcome illustrated how direct scaling approaches can, at times, outperform more complex engineering solutions. === Drawbacks === Reasoning models require significantly more computational resources during inference compared to non-reasoning models. Research on the American Invitational Mathematics Examination (AIME) benchmark found that reasoning models were 10 to 74 times more expensive to operate than their non-reasoning counterparts. The extended inference time is attributed to the detailed, step-by-step reasoning outputs that these models generate, which are typically much longer than responses from standard large language models that provide direct answers without showing their reasoning process. One researcher in early 2025 argued that these models may face potential additional denial-of-service concerns with "overthinking attacks." === Releases === ==== 2024 ==== In September 2024, OpenAI released o1-preview, a large language model with enhanced reasoning capabilities. The full version, o1, was released in December 2024. OpenAI initially shared preliminary results on its successor model, o3, in December 2024, with the full o3 model becoming available in 2025. Alibaba released reasoning versions of its Qwen large language models in November 2024. In December 2024, the company introduced QvQ-72B-Preview, an experimental visual reasoning model. In December 2024, Google introduced Deep Research in Gemini, a feature designed to conduct multi-step research tasks. On December 16, 2024, researchers demonstrated that by scaling test-time compute, a relatively small Llama 3B model could outperform a much larger Llama 70B model on challenging reasoning tasks. This experiment suggested that improved inference strategies can unlock reasoning capabilities even in smaller models. ==== 2025 ==== In January 2025, DeepSeek released R1, a reasoning model that achieved performance comparable to OpenAI's o1 at significantly lower computational cost. The release demonstrated the effectiveness of Group Relative Policy Optimization (GRPO), a reinforcement learning technique used to train the model. On January 25, 2025, DeepSeek enhanced R1 with web search capabilities, allowing the model to retrieve information from the internet while performing reasoning tasks. Research during this period further validated the effectiveness of knowledge distillation for creating reasoning models. The s1-32B model achieved strong performance through budget forcing and scaling methods, reinforcing findings that simpler training approaches can be highly effective for reasoning capabilities. On February 2, 2025, OpenAI released Deep Research, a feature powered by their o3 model that enables users to conduct comprehensive research tasks. The system generates detailed reports by automatically gathering and synthesizing information from multiple web sources. OpenAI called GPT-4.5 its "last non-chain-of-thought model", and implemented with GPT-5 a router model that selects a model based on the difficulty of the task. ==== 2026 ==== In January 2026, Moonshot AI released Kimi K2.5, an open-source 1 trillion parameter MoE model with 32 billion active parameters. It uses an “Agent Swarm” system that dynamically decomposes tasks into sub-agents for reasoning and execution, enabling more scalable multi-step problem solving than a single sequential reasoning chain. == Training == Reasoning models follow the familiar large-scale pretraining used for frontier language models, then diverge in the post-training and optimization. OpenAI reports that o1 is trained with a large-
Sparse dictionary learning
Sparse dictionary learning (also known as sparse coding or SDL) is a representation learning method which aims to find a sparse representation of the input data in the form of a linear combination of basic elements as well as those basic elements themselves. These elements are called atoms, and they compose a dictionary. Atoms in the dictionary are not required to be orthogonal, and they may be an over-complete spanning set. This problem setup also allows the dimensionality of the signals being represented to be higher than any one of the signals being observed. These two properties lead to having seemingly redundant atoms that allow multiple representations of the same signal, but also provide an improvement in sparsity and flexibility of the representation. One of the most important applications of sparse dictionary learning is in the field of compressed sensing or signal recovery. In compressed sensing, a high-dimensional signal can be recovered with only a few linear measurements, provided that the signal is sparse or near-sparse. Since not all signals satisfy this condition, it is crucial to find a sparse representation of that signal such as the wavelet transform or the directional gradient of a rasterized matrix. Once a matrix or a high-dimensional vector is transferred to a sparse space, different recovery algorithms like basis pursuit, CoSaMP, or fast non-iterative algorithms can be used to recover the signal. One of the key principles of dictionary learning is that the dictionary has to be inferred from the input data. The emergence of sparse dictionary learning methods was stimulated by the fact that in signal processing, one typically wants to represent the input data using a minimal amount of components. Before this approach, the general practice was to use predefined dictionaries such as Fourier or wavelet transforms. However, in certain cases, a dictionary that is trained to fit the input data can significantly improve the sparsity, which has applications in data decomposition, compression, and analysis, and has been used in the fields of image denoising and classification, and video and audio processing. Sparsity and overcomplete dictionaries have immense applications in image compression, image fusion, and inpainting. == Problem statement == Given the input dataset X = [ x 1 , . . . , x K ] , x i ∈ R d {\displaystyle X=[x_{1},...,x_{K}],x_{i}\in \mathbb {R} ^{d}} we wish to find a dictionary D ∈ R d × n : D = [ d 1 , . . . , d n ] {\displaystyle \mathbf {D} \in \mathbb {R} ^{d\times n}:D=[d_{1},...,d_{n}]} and a representation R = [ r 1 , . . . , r K ] , r i ∈ R n {\displaystyle R=[r_{1},...,r_{K}],r_{i}\in \mathbb {R} ^{n}} such that both ‖ X − D R ‖ F 2 {\displaystyle \|X-\mathbf {D} R\|_{F}^{2}} is minimized and the representations r i {\displaystyle r_{i}} are sparse enough. This can be formulated as the following optimization problem: argmin D ∈ C , r i ∈ R n ∑ i = 1 K ‖ x i − D r i ‖ 2 2 + λ ‖ r i ‖ 0 {\displaystyle {\underset {\mathbf {D} \in {\mathcal {C}},r_{i}\in \mathbb {R} ^{n}}{\text{argmin}}}\sum _{i=1}^{K}\|x_{i}-\mathbf {D} r_{i}\|_{2}^{2}+\lambda \|r_{i}\|_{0}} , where C ≡ { D ∈ R d × n : ‖ d i ‖ 2 ≤ 1 ∀ i = 1 , . . . , n } {\displaystyle {\mathcal {C}}\equiv \{\mathbf {D} \in \mathbb {R} ^{d\times n}:\|d_{i}\|_{2}\leq 1\,\,\forall i=1,...,n\}} , λ > 0 {\displaystyle \lambda >0} C {\displaystyle {\mathcal {C}}} is required to constrain D {\displaystyle \mathbf {D} } so that its atoms would not reach arbitrarily high values allowing for arbitrarily low (but non-zero) values of r i {\displaystyle r_{i}} . λ {\displaystyle \lambda } controls the trade off between the sparsity and the minimization error. The minimization problem above is not convex because of the ℓ0-"norm" and solving this problem is NP-hard. In some cases L1-norm is known to ensure sparsity and so the above becomes a convex optimization problem with respect to each of the variables D {\displaystyle \mathbf {D} } and R {\displaystyle \mathbf {R} } when the other one is fixed, but it is not jointly convex in ( D , R ) {\displaystyle (\mathbf {D} ,\mathbf {R} )} . === Properties of the dictionary === The dictionary D {\displaystyle \mathbf {D} } defined above can be "undercomplete" if n < d {\displaystyle n