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Implements direct computation of probability density, sampling algorithms, and geometric properties through standardized MCP tool bindings that Claude and other MCP clients can invoke.","intents":["I want to sample from high-dimensional Gaussian distributions and analyze their geometric properties programmatically","I need to demonstrate or verify that high-dimensional Gaussians concentrate on spherical shells","I want to integrate mathematical sampling capabilities into my Claude-based AI workflow without writing custom Python code","I need to compute statistical properties of multivariate Gaussians for research or educational purposes"],"best_for":["researchers and educators demonstrating high-dimensional geometry concepts","AI builders integrating mathematical sampling into Claude-powered applications","data scientists prototyping statistical analyses within Claude conversations","students learning about concentration of measure in high dimensions"],"limitations":["Limited to Gaussian distributions — no support for other probability distributions","Sampling performance degrades significantly beyond ~1000 dimensions due to computational complexity","No built-in visualization — requires external plotting tools to render distributions","MCP protocol adds network round-trip latency for each sampling or analysis operation","No persistence layer — results are ephemeral unless explicitly saved by the client"],"requires":["MCP-compatible client (Claude, or custom MCP client implementation)","Python 3.8+ runtime for the MCP server","NumPy library for numerical computation","MCP SDK or compatible server framework"],"input_types":["integer (dimension count)","float (mean values, covariance parameters)","integer (sample count)","structured JSON (distribution parameters)"],"output_types":["float arrays (samples from distribution)","float (probability density values)","float (geometric properties: radius, concentration metrics)","structured JSON (statistical summaries)"],"categories":["data-processing-analysis","tool-use-integration","mathematical-computing"],"confidence":0.5,"matches":0,"success_rate":0},{"id":"smithery_ae14watanabe-high-dimensional-gaussian-is-like-sphere__cap_1","uri":"capability://data.processing.analysis.gaussian.sphere.concentration.property.computation","name":"gaussian sphere concentration property computation","description":"Computes and demonstrates the mathematical property that high-dimensional Gaussian distributions concentrate their probability mass on spherical shells rather than filling the entire space. 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Implements efficient sampling via Cholesky decomposition of the covariance matrix, enabling generation of correlated high-dimensional samples. Supports both diagonal (independent dimensions) and full covariance structures, with validation of covariance matrix positive-definiteness before sampling.","intents":["I want to generate synthetic data from a multivariate Gaussian with specific correlation structure","I need to sample from a Gaussian with a given covariance matrix for simulation or testing","I want to create correlated random variables for Monte Carlo experiments","I need to validate that my covariance matrix is valid before using it in downstream computations"],"best_for":["statisticians and data scientists running Monte Carlo simulations","machine learning engineers generating synthetic training data with known statistical properties","researchers testing algorithms on Gaussian data with controlled correlation structures","practitioners building probabilistic models that require sampling from Gaussians"],"limitations":["Cholesky decomposition fails for singular or near-singular covariance matrices — requires regularization","Sampling performance is O(d²) due to Cholesky decomposition, limiting practical use beyond ~5000 dimensions","No support for constrained sampling (e.g., samples within a bounded region)","Numerical precision issues arise when covariance matrix condition number exceeds ~10^15"],"requires":["Python 3.8+","NumPy with LAPACK backend for Cholesky decomposition","MCP server runtime"],"input_types":["float array (mean vector, shape d)","float array (covariance matrix, shape d×d)","integer (number of samples to generate)"],"output_types":["float array (samples, shape n×d)","boolean (covariance validity flag)","string (error message if covariance is invalid)"],"categories":["data-processing-analysis","tool-use-integration"],"confidence":0.5,"matches":0,"success_rate":0},{"id":"smithery_ae14watanabe-high-dimensional-gaussian-is-like-sphere__cap_3","uri":"capability://data.processing.analysis.probability.density.evaluation.for.high.dimensional.gaussians","name":"probability density evaluation for high-dimensional gaussians","description":"Computes the probability density function (PDF) values for points in a high-dimensional Gaussian distribution. Implements numerically stable log-PDF computation to avoid underflow in high dimensions, using the formula: log p(x) = -0.5 * (log(det(Σ)) + (x-μ)ᵀΣ⁻¹(x-μ)) + constant. Supports both dense and sparse covariance matrices, with efficient matrix inversion via Cholesky or LU decomposition.","intents":["I want to evaluate the likelihood of data points under a Gaussian model","I need to compute log-probabilities for numerical stability in high dimensions","I want to identify outliers by computing PDF values across a dataset","I need to compute the normalization constant for a Gaussian distribution"],"best_for":["machine learning practitioners computing likelihoods for Gaussian mixture models or VAEs","statisticians performing likelihood-based inference and model comparison","anomaly detection engineers scoring data points against Gaussian models","researchers analyzing probability landscapes in high-dimensional spaces"],"limitations":["Log-PDF computation requires covariance matrix inversion, which is O(d³) and numerically unstable for ill-conditioned matrices","No support for sparse covariance matrices beyond diagonal approximations","Underflow can still occur for extremely low-probability regions (log p < -1000)","Requires storing and inverting the full d×d covariance matrix, limiting scalability beyond ~5000 dimensions"],"requires":["Python 3.8+","NumPy with linear algebra routines","SciPy for special functions (log-determinant computation)","MCP server runtime"],"input_types":["float array (data point or batch of points, shape n×d)","float array (mean vector, shape d)","float array (covariance matrix, shape d×d)"],"output_types":["float (log-PDF value for single point)","float array (log-PDF values for batch, shape n)","float (PDF value in original probability space, if requested)"],"categories":["data-processing-analysis","tool-use-integration"],"confidence":0.5,"matches":0,"success_rate":0},{"id":"smithery_ae14watanabe-high-dimensional-gaussian-is-like-sphere__cap_4","uri":"capability://data.processing.analysis.dimension.aware.geometric.property.analysis","name":"dimension-aware geometric property analysis","description":"Analyzes how geometric properties of Gaussian distributions change with dimensionality, computing metrics such as expected distance from origin, volume of probability mass shells, and the ratio of surface area to volume in high-dimensional spaces. Provides both theoretical predictions (via chi-squared distribution properties) and empirical measurements from samples, enabling comparison of theory vs. practice across dimensions.","intents":["I want to understand how the expected distance from the origin scales with dimension","I need to compute the volume of probability mass at different radii in high dimensions","I want to verify the curse of dimensionality through concrete geometric metrics","I need to understand how probability concentrates on lower-dimensional manifolds as dimension increases"],"best_for":["educators teaching high-dimensional geometry and concentration of measure","researchers studying manifold learning and dimensionality reduction","machine learning engineers building dimension-aware algorithms","practitioners analyzing neural network representations in high-dimensional spaces"],"limitations":["Theoretical predictions assume isotropic Gaussians — anisotropic cases require numerical integration","Empirical measurements require large sample sizes (>10,000) for accurate estimates in dimensions > 100","No support for computing properties on constrained manifolds or subspaces","Computational cost grows as O(d) for theoretical predictions and O(n*d) for empirical measurements"],"requires":["Python 3.8+","NumPy and SciPy for chi-squared distribution and special functions","MCP server runtime"],"input_types":["integer (dimension count, d)","integer (number of samples for empirical measurement)","float (standard deviation of Gaussian)"],"output_types":["float (expected distance from origin)","float (standard deviation of distance)","float array (probability mass at different radii)","structured JSON (comprehensive geometric summary)"],"categories":["data-processing-analysis","planning-reasoning"],"confidence":0.5,"matches":0,"success_rate":0}],"trust":{"score":26,"verified":false,"data_access_risk":"moderate","permissions":["MCP-compatible client (Claude, or custom MCP client implementation)","Python 3.8+ runtime for the MCP server","NumPy library for numerical computation","MCP SDK or compatible server framework","Python 3.8+","NumPy for linear algebra and numerical integration","SciPy for special functions (gamma, chi-squared distributions)","MCP server runtime","NumPy with LAPACK backend for Cholesky decomposition","NumPy with linear algebra routines"],"failure_modes":["Limited to Gaussian distributions — no support for other probability distributions","Sampling performance degrades significantly beyond ~1000 dimensions due to computational complexity","No built-in visualization — requires external plotting tools to render distributions","MCP protocol adds network round-trip latency for each sampling or analysis operation","No persistence layer — results are ephemeral unless explicitly saved by the client","Analytical formulas become numerically unstable for dimensions > 10,000","Concentration metrics assume isotropic Gaussians — anisotropic covariances require custom computation","No support for computing concentration in constrained subspaces or manifolds","Requires sufficient samples for empirical verification — theoretical bounds may not match small-sample behavior","Cholesky decomposition fails for singular or near-singular covariance matrices — requires regularization","builder identity is not verified yet","no observed match outcomes yet"],"rank_breakdown":{"adoption":0.05,"quality":0.2,"ecosystem":0.48999999999999994,"match_graph":0.25,"freshness":0.52,"weights":{"adoption":0.25,"quality":0.25,"ecosystem":0.15,"match_graph":0.23,"freshness":0.12}},"observed_outcomes":{"matches":0,"success_rate":0,"avg_confidence":0,"top_intents":[],"last_matched_at":null},"maintenance":{"status":"active","updated_at":"2026-05-24T12:16:25.062Z","last_scraped_at":"2026-05-03T15:19:16.961Z","last_commit":null},"community":{"stars":null,"forks":null,"weekly_downloads":null,"model_downloads":null,"model_likes":null}},"distribution":{"claim_url":"https://unfragile.ai/submit?claim=ae14watanabe-high-dimensional-gaussian-is-like-sphere","compare_url":"https://unfragile.ai/compare?artifact=ae14watanabe-high-dimensional-gaussian-is-like-sphere"}},"signature":"bTivb6xLICcUH594R1BQzsn7k35NIGF1uI4KtJyOPe8TwXfGTjA5b8112kFpJSWgHfiVTSw1VouqASzJzNxIAA==","signedAt":"2026-07-08T07:45:37.919Z","signedBy":"unfragile.ai","version":1},"_links":{"self":"https://unfragile.ai/api/v1/passport/ae14watanabe-high-dimensional-gaussian-is-like-sphere","artifact":"https://unfragile.ai/ae14watanabe-high-dimensional-gaussian-is-like-sphere","verify":"https://unfragile.ai/api/v1/verify?slug=ae14watanabe-high-dimensional-gaussian-is-like-sphere","publicKey":"https://unfragile.ai/api/v1/trust-passport-public-key","spec":"https://unfragile.ai/trust","schema":"https://unfragile.ai/schema.json","docs":"https://unfragile.ai/docs"}}