Stories with Tag coding theory

• Cross-Entropy and KL Divergence

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  Posted: 2025-04-13 04:48:48

  Verbose tl;dr

  Cross-entropy and KL divergence are closely related measures of difference between probability distributions. While cross-entropy quantifies the average number of bits needed to encode events drawn from a true distribution p using a coding scheme optimized for a predicted distribution q, KL divergence measures how much more information is needed on average when using q instead of p. Specifically, KL divergence is the difference between cross-entropy and the entropy of the true distribution p. Therefore, minimizing cross-entropy with respect to q is equivalent to minimizing the KL divergence, as the entropy of p is constant. While both can measure the dissimilarity between distributions, KL divergence is a true "distance" metric (though asymmetric), whereas cross-entropy is not. The post illustrates these concepts with detailed numerical examples and explains their significance in machine learning, particularly for tasks like classification where the goal is to match a predicted distribution to the true data distribution.

  This blog post delves into the relationship between cross-entropy and Kullback-Leibler (KL) divergence, two important concepts in information theory and machine learning, particularly within the context of classification problems. It begins by laying a foundation by defining entropy, which quantifies the average amount of information needed to represent an event drawn from a probability distribution. A lower entropy indicates less uncertainty, meaning the distribution is more predictable.

  The post then progresses to cross-entropy, explaining that it measures the average number of bits required to encode an event drawn from a true probability distribution, p, using a coding scheme optimized for a different, predicted probability distribution, q. Essentially, it quantifies the inefficiency introduced when using a suboptimal coding scheme based on an incorrect prediction of the true distribution. A lower cross-entropy implies a better alignment between the predicted and true distributions.

  The core of the post lies in elucidating the connection between cross-entropy and KL divergence. KL divergence, also known as relative entropy, measures how different one probability distribution is from a second, reference probability distribution. In other words, it quantifies the information lost when using one distribution to approximate another. The post meticulously demonstrates mathematically that the cross-entropy between p and q can be decomposed into two terms: the entropy of the true distribution, p, and the KL divergence between p and q.

  This decomposition is crucial because it reveals why minimizing cross-entropy in machine learning is equivalent to minimizing the KL divergence between the predicted and true distributions. Since the entropy of the true distribution is a constant, unaffected by our predictions, any reduction in cross-entropy directly translates to a reduction in KL divergence, meaning our predictions are becoming more accurate representations of the true distribution.

  The post uses a concrete example with a simple two-class classification problem to illustrate these concepts. It shows how calculating the cross-entropy and KL divergence provides insights into the performance of a classifier. Furthermore, it highlights that optimizing a classification model by minimizing cross-entropy effectively amounts to minimizing the information lost when approximating the true label distribution with the predicted probabilities.

  In summary, the post provides a comprehensive explanation of cross-entropy and KL divergence, clearly outlining their definitions, mathematical relationship, and significance in machine learning. It emphasizes the practical implication that minimizing cross-entropy during training leads to more accurate predictions by effectively minimizing the difference between the predicted and true data distributions. The post concludes by reiterating the importance of understanding these concepts for anyone working with machine learning models, especially in classification tasks.

  • Cross-Entropy
  • KL Divergence
  • information theory
  • probability
  • Statistics
  • machine learning
  • deep learning
  • Loss Function
  • optimization
  • entropy
  • Information Gain
  • Relative Entropy
  • coding theory
  • Data Science

  Summary of Comments ( 4 )
  https://news.ycombinator.com/item?id=43670171

  Verbose tl;dr

  Hacker News users generally praised the clarity and helpfulness of the article explaining cross-entropy and KL divergence. Several commenters pointed out the value of the concrete code examples and visualizations provided. One user appreciated the explanation of the difference between minimizing cross-entropy and maximizing likelihood, while another highlighted the article's effective use of simple language to explain complex concepts. A few comments focused on practical applications, including how cross-entropy helps in model selection and its relation to log loss. Some users shared additional resources and alternative explanations, further enriching the discussion.

  The Hacker News post titled "Cross-Entropy and KL Divergence," linking to an article explaining these concepts, has generated several comments. Many commenters appreciate the clarity and helpfulness of the article.

  One commenter points out a potential area of confusion in the article regarding the base of the logarithm used in the calculations. They explain that while the article uses base 2 for its examples, other bases like e (natural logarithm) are common, and the choice affects the units (bits vs. nats) of the result. This commenter emphasizes the importance of understanding the relationship between these different units and how the chosen base impacts the interpretation of the calculated values.

  Another commenter expresses gratitude for the clear and concise explanation, stating that they've often seen these terms used without proper definition. They specifically praise the article's use of concrete examples and its intuitive approach to explaining complex mathematical concepts.

  Another comment focuses on the practical implications of cross-entropy, particularly its use in machine learning as a loss function. They discuss how minimizing cross-entropy leads to improved model performance and how it relates to maximizing the likelihood of the observed data. This comment connects the theoretical concepts to real-world applications, enhancing the practical understanding of the topic.

  One user provides a link to another resource, a blog post by Tim Vieira, which offers further explanation and builds upon the original article's content. This contribution extends the discussion by providing additional avenues for learning and exploring related concepts.

  A few other commenters express their agreement with the positive sentiment towards the article, confirming its usefulness and clarity. They appreciate the article's straightforward approach and the way it demystifies these often-confusing concepts.

  In summary, the comments on the Hacker News post overwhelmingly praise the linked article for its clear and accessible explanation of cross-entropy and KL divergence. They delve into specific aspects like the importance of the logarithm base, the practical applications in machine learning, and provide additional resources for further learning. The comments contribute to a deeper understanding and appreciation of the article's subject matter.

• Succinct Data Structures

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  Posted: 2025-03-06 17:48:37

  Verbose tl;dr

  Succinct data structures represent data in space close to the information-theoretic lower bound, while still allowing efficient queries. The blog post explores several examples, starting with representing a bit vector using only one extra bit beyond the raw data, while still supporting constant-time rank and select operations. It then extends this to compressed bit vectors using Elias-Fano encoding and explains how to represent arbitrary sets and sparse arrays succinctly. Finally, it touches on representing trees succinctly, demonstrating how to support various navigation operations efficiently despite the compact representation. Overall, the post emphasizes the power of succinct data structures to achieve substantial space savings without significant performance degradation.

  The blog post "Succinct Data Structures" delves into the fascinating realm of representing data structures in a manner that approaches the information-theoretic lower bound of space complexity while still permitting efficient query operations. This means storing data using close to the minimum number of bits theoretically required to represent the information, without sacrificing the speed of accessing and using that data.

  The author begins by establishing the fundamental concept of information-theoretic lower bounds. This refers to the absolute minimum number of bits needed to differentiate between all possible configurations of a data structure. For example, representing a bit vector of length n requires, at minimum, n bits, while a permutation of n elements necessitates approximately n log n bits (using logarithms base 2). These lower bounds provide a benchmark against which the efficiency of succinct data structures can be measured.

  The post then introduces several classic examples of succinct data structures, beginning with Elias-Fano encoding. This technique efficiently represents a monotonically increasing sequence of integers, a common scenario in various applications. The key idea behind Elias-Fano is to separate the binary representation of each integer into high and low bits, storing them in separate structures optimized for their respective characteristics. This allows for efficient rank and select operations, which are fundamental to many algorithms operating on such sequences.

  The discussion continues with the representation of bit vectors. While storing a bit vector trivially uses n bits, succinct representations aim to support operations like rank (counting the number of set bits up to a given position) and select (finding the position of the k-th set bit) efficiently within a space very close to n bits. These representations often employ ingenious techniques like blocking and precomputed tables to achieve constant-time or near constant-time query operations.

  Next, the post touches upon succinct tree representations. Representing a tree efficiently while supporting navigation operations is crucial in many applications. Several succinct tree representations are mentioned, each using different strategies to encode the tree structure and enable operations like finding the parent, children, or subtree size of a node. These techniques often involve clever bit manipulations and carefully designed auxiliary structures.

  The author emphasizes the importance of operations like rank and select in navigating and utilizing these succinct data structures. These functions become the building blocks for higher-level operations, allowing for efficient querying and manipulation of the underlying data despite its compressed representation.

  Finally, the post briefly discusses practical considerations related to succinct data structures. While achieving theoretical optimality in terms of space is a primary goal, the constant factors associated with the complexities of these structures can impact their practical performance. The author concludes by noting the continuing research and development in this area, suggesting the potential for even more efficient and versatile succinct data structures in the future. The post serves as an excellent introduction to the fundamental concepts and techniques of succinct data structures, illustrating their power and utility in representing large datasets efficiently.

  • data structures
  • succinct data structures
  • algorithms
  • data compression
  • bit manipulation
  • space efficiency
  • memory efficiency
  • information theory
  • coding theory
  • Computer Science

  Summary of Comments ( 27 )
  https://news.ycombinator.com/item?id=43282995

  Verbose tl;dr

  Hacker News users discussed the practicality and performance trade-offs of succinct data structures. Some questioned the real-world benefits given the complexity and potential performance hits compared to simpler, less space-efficient solutions, especially with the abundance of cheap memory. Others highlighted the value in specific niches like bioinformatics and embedded systems where memory is constrained. The discussion also touched on the difficulty of implementing and debugging these structures and the lack of mature libraries in common languages. A compelling comment highlighted the use case of storing large language models efficiently, where succinct data structures can significantly reduce storage requirements and memory access times, potentially enabling new applications on resource-constrained devices. Others noted the theoretical elegance of the approach, even if practical applications remain somewhat niche.

  The Hacker News post "Succinct Data Structures" spawned a moderately active discussion with a mix of practical observations, theoretical considerations, and personal anecdotes.

  Several commenters focused on the practical applications, or lack thereof, of succinct data structures. One commenter questioned the real-world utility outside of specialized domains like bioinformatics, expressing skepticism about their general applicability due to the complexity and constant factors involved. Another agreed, pointing out that the performance gains are often marginal and not worth the added code complexity in most cases. A counterpoint was raised by someone who suggested potential benefits for embedded systems or scenarios with extremely tight memory constraints.

  The discussion also delved into the theoretical aspects of succinctness. One commenter highlighted the connection between succinct data structures and information theory, noting how they push the boundaries of representing data with minimal overhead. Another brought up the trade-off between succinctness and query time, emphasizing that achieving extreme compression often comes at the cost of slower access speeds.

  A few commenters shared their personal experiences and preferences. One admitted finding the concepts fascinating but acknowledged the limited practical use in their day-to-day work. Another expressed a preference for simpler data structures that prioritize readability and maintainability over marginal performance gains.

  A couple of comments also touched on specific data structure implementations. One commenter mentioned Elias-Fano coding as a particularly useful technique for representing sorted sets, while another brought up wavelet trees and their applications in compressed string indexing.

  Overall, the comments reflect a nuanced view of succinct data structures. While acknowledging their theoretical elegance and potential benefits in specific niches, many commenters expressed reservations about their widespread adoption due to complexity and limited practical gains in common scenarios. The discussion highlights the importance of carefully considering the trade-offs between space efficiency, performance, and code complexity when choosing data structures.

• Iterated Log Coding

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  Posted: 2025-02-26 07:43:21

  Verbose tl;dr

  Iterated Log Coding (ILC) offers a novel approach to data compression by representing integers as a series of logarithmic operations. Instead of traditional methods like Huffman coding or arithmetic coding, ILC leverages the repeated application of the logarithm to achieve potentially superior compression for certain data distributions. It encodes an integer by counting how many times the logarithm base b needs to be applied before the result falls below a threshold. This "iteration count" becomes the core of the compressed representation, supplemented by a fractional value representing the remainder after the final logarithm application. Decoding reverses this process, effectively "exponentiating" the iteration count and incorporating the fractional remainder. While the blog post acknowledges that ILC's practical usefulness requires further investigation, it highlights the theoretical potential and presents a basic implementation in Python.

  This blog post introduces a novel compression technique called "Iterated Log Coding," or "iterlog," which cleverly exploits the predictable nature of sorted integer sequences. The core idea revolves around representing each integer in a sorted sequence not by its absolute value, but by the difference between it and the preceding integer, a concept similar to delta encoding. However, iterlog takes this further by recursively applying this differencing process. Instead of just storing these differences directly, it stores the base-2 logarithm of each difference, rounded up to the nearest integer. This process of taking the log of the differences is iterated until all remaining values become zero.

  The author argues that this approach is particularly well-suited for compressing sorted sequences containing clusters of similar values, a common characteristic of many real-world datasets. When integers are close together, their differences, and consequently the logarithms of these differences, will be small. Representing these small values requires fewer bits, leading to significant compression. The iterated nature of the algorithm further amplifies this effect. As the differences shrink through successive iterations, the logarithms also decrease, leading to progressively smaller representations.

  The blog post details the encoding and decoding processes with illustrative examples. Encoding involves repeatedly calculating differences between consecutive numbers and then their logarithms, storing the rounded-up logarithm values at each level. A key point is that the first number in the sequence and the number of elements in the sequence need to be stored separately as they are not part of the differencing process. The number of iterations required is also implicitly stored by the presence of all zeros in the final iteration. Decoding reverses this process by repeatedly exponentiating and summing the logged differences, effectively reconstructing the original sequence.

  The author acknowledges that iterlog coding is not a general-purpose compression algorithm and might not be suitable for all types of data. It's particularly effective for sorted sequences with clustered values, where the differences between successive elements are small. In situations where the differences are large and vary significantly, iterlog may not offer significant compression advantages or might even increase the data size compared to uncompressed representation. The post concludes with a Python implementation of the iterlog encoding and decoding algorithms, allowing readers to experiment with the technique and evaluate its performance on different datasets. The author invites further exploration and optimization of the idea, suggesting that varying the logarithm base or utilizing alternative rounding strategies could potentially improve compression ratios in specific scenarios.

  • data compression
  • iterated logarithm
  • logarithm
  • coding theory
  • compression algorithm
  • binary coding
  • Elias coding
  • variable-length coding
  • integer compression
  • succinct data structures

  Summary of Comments ( 23 )
  https://news.ycombinator.com/item?id=43181610

  Verbose tl;dr

  Hacker News users generally praised the clarity and novelty of the Iterated Log Coding approach. Several commenters appreciated the author's clear explanation of a complex topic and the potential benefits of the technique for compression, especially in specialized domains like bioinformatics. Some discussed its similarities to Huffman coding and Elias gamma coding, suggesting it falls within a family of variable-length codes optimized for certain data distributions. A few pointed out limitations or offered alternative implementations, including using a lookup table for smaller values of 'n' for performance improvements. The practicality of the method for general-purpose compression was questioned, with some suggesting it might be too niche, while others found it theoretically interesting and a valuable addition to existing compression methods.

  The Hacker News post "Iterated Log Coding" discussing the blog post about a new compression algorithm has generated a moderate amount of discussion, with several commenters engaging with the core ideas presented.

  One of the most compelling threads revolves around the practicality and novelty of the "iterated log" approach. A user points out that the method is reminiscent of Elias gamma coding, a well-established variable-length coding scheme. This sparked further discussion comparing the two methods, with some suggesting that iterated log coding might offer advantages in certain scenarios, particularly when dealing with very large numbers, while others remain skeptical, highlighting the efficiency and existing implementations of Elias gamma coding.

  Another commenter questions the choice of using base-10 logarithms in the examples, suggesting that base-2 logarithms would be more natural in a computational context. This comment prompts a brief discussion about the rationale behind the base choice, with the possibility that base-10 was selected for easier human readability in the illustrative examples.

  Several commenters express interest in seeing benchmarks and comparisons against existing compression algorithms. They emphasize the importance of real-world performance data to evaluate the effectiveness of the proposed method. One user specifically asks about the performance on typical integer sequences found in practice.

  There's also a short exchange regarding the potential applications of this compression method. One commenter suggests it could be useful for compressing indexes in databases.

  Finally, a few commenters delve into the theoretical underpinnings of the algorithm, discussing its relationship to other coding schemes and exploring its potential limitations. One user mentions the connection to prefix codes and how the unique decodability property is ensured.

  Overall, the comments section reveals a mixture of curiosity, skepticism, and cautious optimism towards the iterated log coding approach. While some see it as a potentially interesting idea with specific niche applications, others remain unconvinced of its practical value compared to established techniques. The prevailing sentiment appears to be a desire for more empirical evidence and comparisons to solidify the claims made in the original blog post.

• An alternative construction of Shannon entropy

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  Posted: 2024-11-13 16:45:13

  Verbose tl;dr

  This blog post presents a different way to derive Shannon entropy, focusing on its property as a unique measure of information content. Instead of starting with desired properties like additivity and then finding a formula that satisfies them, the author begins with a core idea: measuring the average number of binary questions needed to pinpoint a specific outcome from a probability distribution. By formalizing this concept using a binary tree representation of the questioning process and leveraging Kraft's inequality, they demonstrate that -∑pᵢlog₂(pᵢ) emerges naturally as the optimal average question length, thus establishing it as the entropy. This construction emphasizes the intuitive link between entropy and the efficient encoding of information.

  This blog post presents a different perspective on deriving Shannon entropy, distinct from the traditional axiomatic approach. Instead of starting with desired properties and deducing the entropy formula, it begins with a fundamental problem: quantifying the average number of bits needed to optimally represent outcomes from a probabilistic source. The author argues this approach provides a more intuitive and grounded understanding of why the entropy formula takes the shape it does.

  The post meticulously constructs this derivation. It starts by considering a source emitting symbols from a finite alphabet, each with an associated probability. The core idea is to group these symbols into sets based on their probabilities, specifically targeting sets where the cumulative probability is a power of two. This allows for efficient representation using binary codes, as each set can be uniquely identified by a binary prefix.

  The process begins with the most probable symbol and continues iteratively, grouping less probable symbols into progressively larger sets until all symbols are assigned. The author demonstrates how this grouping mirrors the process of building a Huffman code, a well-known algorithm for creating optimal prefix-free codes.

  The post then carefully analyzes the expected number of bits required to encode a symbol using this method. This expectation involves summing the product of the number of bits assigned to a set (which relates to the negative logarithm of the cumulative probability of that set) and the cumulative probability of the symbols within that set.

  Through a series of mathematical manipulations and approximations, leveraging the properties of logarithms and the behavior of probabilities as the number of samples increases, the author shows that this expected number of bits converges to the familiar Shannon entropy formula: the negative sum of each symbol's probability multiplied by the logarithm base 2 of that probability.

  Crucially, the derivation highlights the relationship between optimal coding and entropy. It demonstrates that Shannon entropy represents the theoretical lower bound on the average number of bits needed to encode messages from a given source, achievable through optimal coding schemes like Huffman coding. This construction emphasizes that entropy is not just a measure of uncertainty or information content, but intrinsically linked to efficient data compression and representation. The post concludes by suggesting this alternative construction offers a more concrete and less abstract understanding of Shannon entropy's significance in information theory.

  • information theory
  • Shannon entropy
  • entropy
  • mathematics
  • probability
  • coding theory
  • data compression
  • alternative construction
  • axiomatic definition
  • foundations of information theory
  • Statistics
  • Information Measurement
  • axiomatic systems
  • axiomatic derivation

  Summary of Comments ( 11 )
  https://news.ycombinator.com/item?id=42127609

  Verbose tl;dr

  Hacker News users discuss the alternative construction of Shannon entropy presented in the linked article. Some express appreciation for the clear explanation and visualizations, finding the geometric approach insightful and offering a fresh perspective on a familiar concept. Others debate the pedagogical value of the approach, questioning whether it truly simplifies understanding for those unfamiliar with entropy, or merely offers a different lens for those already versed in the subject. A few commenters note the connection to cross-entropy and Kullback-Leibler divergence, suggesting the geometric interpretation could be extended to these related concepts. There's also a brief discussion on the practical implications and potential applications of this alternative construction, although no concrete examples are provided. Overall, the comments reflect a mix of appreciation for the novel approach and a pragmatic assessment of its usefulness in teaching and application.

  The Hacker News post titled "An alternative construction of Shannon entropy," linking to an article exploring a different way to derive Shannon entropy, has generated a moderate discussion with several interesting comments.

  One commenter highlights the pedagogical value of the approach presented in the article. They appreciate how it starts with desirable properties for a measure of information and derives the entropy formula from those, contrasting this with the more common axiomatic approach where the formula is presented and then shown to satisfy the properties. They believe this method makes the concept of entropy more intuitive.

  Another commenter focuses on the historical context, mentioning that Shannon's original derivation was indeed based on desired properties. They point out that the article's approach is similar to the one Shannon employed, further reinforcing the pedagogical benefit of seeing the formula emerge from its intended properties rather than the other way around. They link to a relevant page within a book on information theory which seemingly discusses Shannon's original derivation.

  A third commenter questions the novelty of the approach, suggesting that it seems similar to standard treatments of the topic. They wonder if the author might be overselling the "alternative construction" aspect. This sparks a brief exchange with another user who defends the article, arguing that while the fundamental ideas are indeed standard, the specific presentation and the emphasis on the grouping property could offer a fresh perspective, especially for educational purposes.

  Another commenter delves into more technical details, discussing the concept of entropy as a measure of average code length and relating it to Kraft's inequality. They connect this idea to the article's approach, demonstrating how the desired properties lead to a formula that aligns with the coding interpretation of entropy.

  Finally, a few comments touch upon related concepts like cross-entropy and Kullback-Leibler divergence, briefly extending the discussion beyond the scope of the original article. One commenter mentions an example of how entropy is useful, by stating how optimizing for log-loss in a neural network can be interpreted as an attempt to make the predicted distribution very similar to the true distribution.

  Overall, the comments section provides a valuable supplement to the article, offering different perspectives on its significance, clarifying some technical points, and connecting it to broader concepts in information theory. While not groundbreaking, the discussion reinforces the importance of pedagogical approaches that derive fundamental formulas from their intended properties.