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a37c71153d
...
86fa682b31
app-proto
engine-proto/gram-test
@ -10,7 +10,6 @@ default = ["console_error_panic_hook"]
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[dependencies]
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itertools = "0.13.0"
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js-sys = "0.3.70"
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lazy_static = "1.5.0"
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nalgebra = "0.33.0"
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rustc-hash = "2.0.0"
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slab = "0.4.9"
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@ -1,8 +0,0 @@
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# based on "Enabling print statements in Cargo tests", by Jon Almeida
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#
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# https://jonalmeida.com/posts/2015/01/23/print-cargo/
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#
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cargo test -- --nocapture engine::tests::irisawa_hexlet_test
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cargo test -- --nocapture engine::tests::three_spheres_example
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cargo test -- --nocapture engine::tests::point_on_sphere_example
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@ -12,8 +12,7 @@ fn load_gen_assemb(assembly: &Assembly) {
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label: String::from("Castor"),
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color: [1.00_f32, 0.25_f32, 0.00_f32],
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rep: engine::sphere(0.5, 0.5, 0.0, 1.0),
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constraints: BTreeSet::default(),
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index: 0
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constraints: BTreeSet::default()
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}
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);
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let _ = assembly.try_insert_element(
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@ -22,8 +21,7 @@ fn load_gen_assemb(assembly: &Assembly) {
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label: String::from("Pollux"),
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color: [0.00_f32, 0.25_f32, 1.00_f32],
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rep: engine::sphere(-0.5, -0.5, 0.0, 1.0),
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constraints: BTreeSet::default(),
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index: 0
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constraints: BTreeSet::default()
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}
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);
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let _ = assembly.try_insert_element(
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@ -32,8 +30,7 @@ fn load_gen_assemb(assembly: &Assembly) {
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label: String::from("Ursa major"),
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color: [0.25_f32, 0.00_f32, 1.00_f32],
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rep: engine::sphere(-0.5, 0.5, 0.0, 0.75),
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constraints: BTreeSet::default(),
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index: 0
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constraints: BTreeSet::default()
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}
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);
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let _ = assembly.try_insert_element(
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@ -42,8 +39,7 @@ fn load_gen_assemb(assembly: &Assembly) {
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label: String::from("Ursa minor"),
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color: [0.25_f32, 1.00_f32, 0.00_f32],
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rep: engine::sphere(0.5, -0.5, 0.0, 0.5),
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constraints: BTreeSet::default(),
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index: 0
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constraints: BTreeSet::default()
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}
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);
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let _ = assembly.try_insert_element(
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@ -52,8 +48,7 @@ fn load_gen_assemb(assembly: &Assembly) {
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label: String::from("Deimos"),
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color: [0.75_f32, 0.75_f32, 0.00_f32],
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rep: engine::sphere(0.0, 0.15, 1.0, 0.25),
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constraints: BTreeSet::default(),
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index: 0
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constraints: BTreeSet::default()
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}
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);
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let _ = assembly.try_insert_element(
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@ -62,8 +57,17 @@ fn load_gen_assemb(assembly: &Assembly) {
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label: String::from("Phobos"),
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color: [0.00_f32, 0.75_f32, 0.50_f32],
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rep: engine::sphere(0.0, -0.15, -1.0, 0.25),
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constraints: BTreeSet::default(),
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index: 0
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constraints: BTreeSet::default()
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}
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);
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assembly.insert_constraint(
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Constraint {
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args: (
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assembly.elements_by_id.with_untracked(|elts_by_id| elts_by_id["gemini_a"]),
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assembly.elements_by_id.with_untracked(|elts_by_id| elts_by_id["gemini_b"])
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),
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rep: 0.5,
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active: create_signal(true)
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}
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);
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}
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@ -77,8 +81,7 @@ fn load_low_curv_assemb(assembly: &Assembly) {
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label: "Central".to_string(),
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color: [0.75_f32, 0.75_f32, 0.75_f32],
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rep: engine::sphere(0.0, 0.0, 0.0, 1.0),
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constraints: BTreeSet::default(),
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index: 0
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constraints: BTreeSet::default()
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}
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);
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let _ = assembly.try_insert_element(
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@ -87,8 +90,7 @@ fn load_low_curv_assemb(assembly: &Assembly) {
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label: "Assembly plane".to_string(),
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color: [0.75_f32, 0.75_f32, 0.75_f32],
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rep: engine::sphere_with_offset(0.0, 0.0, 1.0, 0.0, 0.0),
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constraints: BTreeSet::default(),
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index: 0
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constraints: BTreeSet::default()
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}
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);
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let _ = assembly.try_insert_element(
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@ -97,8 +99,7 @@ fn load_low_curv_assemb(assembly: &Assembly) {
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label: "Side 1".to_string(),
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color: [1.00_f32, 0.00_f32, 0.25_f32],
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rep: engine::sphere_with_offset(1.0, 0.0, 0.0, 1.0, 0.0),
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constraints: BTreeSet::default(),
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index: 0
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constraints: BTreeSet::default()
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}
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);
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let _ = assembly.try_insert_element(
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@ -107,8 +108,7 @@ fn load_low_curv_assemb(assembly: &Assembly) {
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label: "Side 2".to_string(),
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color: [0.25_f32, 1.00_f32, 0.00_f32],
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rep: engine::sphere_with_offset(-0.5, a, 0.0, 1.0, 0.0),
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constraints: BTreeSet::default(),
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index: 0
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constraints: BTreeSet::default()
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}
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);
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let _ = assembly.try_insert_element(
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@ -117,8 +117,7 @@ fn load_low_curv_assemb(assembly: &Assembly) {
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label: "Side 3".to_string(),
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color: [0.00_f32, 0.25_f32, 1.00_f32],
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rep: engine::sphere_with_offset(-0.5, -a, 0.0, 1.0, 0.0),
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constraints: BTreeSet::default(),
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index: 0
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constraints: BTreeSet::default()
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}
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);
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let _ = assembly.try_insert_element(
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@ -127,8 +126,7 @@ fn load_low_curv_assemb(assembly: &Assembly) {
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label: "Corner 1".to_string(),
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color: [0.75_f32, 0.75_f32, 0.75_f32],
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rep: engine::sphere(-4.0/3.0, 0.0, 0.0, 1.0/3.0),
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constraints: BTreeSet::default(),
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index: 0
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constraints: BTreeSet::default()
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}
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);
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let _ = assembly.try_insert_element(
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@ -137,8 +135,7 @@ fn load_low_curv_assemb(assembly: &Assembly) {
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label: "Corner 2".to_string(),
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color: [0.75_f32, 0.75_f32, 0.75_f32],
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rep: engine::sphere(2.0/3.0, -4.0/3.0 * a, 0.0, 1.0/3.0),
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constraints: BTreeSet::default(),
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index: 0
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constraints: BTreeSet::default()
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}
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);
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let _ = assembly.try_insert_element(
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@ -147,8 +144,7 @@ fn load_low_curv_assemb(assembly: &Assembly) {
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label: String::from("Corner 3"),
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color: [0.75_f32, 0.75_f32, 0.75_f32],
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rep: engine::sphere(2.0/3.0, 4.0/3.0 * a, 0.0, 1.0/3.0),
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constraints: BTreeSet::default(),
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index: 0
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constraints: BTreeSet::default()
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}
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);
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}
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@ -219,7 +215,6 @@ pub fn AddRemove() -> View {
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rep: 0.0,
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active: create_signal(true)
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});
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state.assembly.realize();
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state.selection.update(|sel| sel.clear());
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/* DEBUG */
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@ -1,11 +1,8 @@
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use nalgebra::{DMatrix, DVector};
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use nalgebra::DVector;
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use rustc_hash::FxHashMap;
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use slab::Slab;
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use std::collections::BTreeSet;
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use sycamore::prelude::*;
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use web_sys::{console, wasm_bindgen::JsValue}; /* DEBUG */
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use crate::engine::{realize_gram, PartialMatrix};
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#[derive(Clone, PartialEq)]
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pub struct Element {
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@ -13,10 +10,7 @@ pub struct Element {
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pub label: String,
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pub color: [f32; 3],
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pub rep: DVector<f64>,
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pub constraints: BTreeSet<usize>,
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// internal properties, not reflected in any view
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pub index: usize
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pub constraints: BTreeSet<usize>
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}
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#[derive(Clone)]
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@ -46,8 +40,6 @@ impl Assembly {
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}
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}
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// --- inserting elements and constraints ---
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// insert an element into the assembly without checking whether we already
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// have an element with the same identifier. any element that does have the
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// same identifier will get kicked out of the `elements_by_id` index
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@ -85,8 +77,7 @@ impl Assembly {
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label: format!("Sphere {}", id_num),
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color: [0.75_f32, 0.75_f32, 0.75_f32],
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rep: DVector::<f64>::from_column_slice(&[0.0, 0.0, 0.0, 0.5, -0.5]),
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constraints: BTreeSet::default(),
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index: 0
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constraints: BTreeSet::default()
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}
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);
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}
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@ -99,83 +90,4 @@ impl Assembly {
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elts[args.1].constraints.insert(key);
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})
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}
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// --- realization ---
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pub fn realize(&self) {
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// index the elements
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self.elements.update_silent(|elts| {
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for (index, (_, elt)) in elts.into_iter().enumerate() {
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elt.index = index;
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}
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});
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// set up the Gram matrix and the initial configuration matrix
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let (gram, guess) = self.elements.with_untracked(|elts| {
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// set up the off-diagonal part of the Gram matrix
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let mut gram_to_be = PartialMatrix::new();
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self.constraints.with_untracked(|csts| {
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for (_, cst) in csts {
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let args = cst.args;
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let row = elts[args.0].index;
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let col = elts[args.1].index;
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gram_to_be.push_sym(row, col, cst.rep);
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}
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});
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// set up the initial configuration matrix and the diagonal of the
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// Gram matrix
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let mut guess_to_be = DMatrix::<f64>::zeros(5, elts.len());
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for (_, elt) in elts {
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let index = elt.index;
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gram_to_be.push_sym(index, index, 1.0);
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guess_to_be.set_column(index, &elt.rep);
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}
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(gram_to_be, guess_to_be)
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});
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/* DEBUG */
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// log the Gram matrix
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console::log_1(&JsValue::from("Gram matrix:"));
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gram.log_to_console();
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/* DEBUG */
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// log the initial configuration matrix
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console::log_1(&JsValue::from("old configuration:"));
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for j in 0..guess.nrows() {
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let mut row_str = String::new();
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for k in 0..guess.ncols() {
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row_str.push_str(format!(" {:>8.3}", guess[(j, k)]).as_str());
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}
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console::log_1(&JsValue::from(row_str));
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}
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// look for a configuration with the given Gram matrix
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let (config, success, history) = realize_gram(
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&gram, guess, &[],
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1.0e-12, 0.5, 0.9, 1.1, 200, 110
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);
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/* DEBUG */
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// report the outcome of the search
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console::log_1(&JsValue::from(
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if success {
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"Target accuracy achieved!"
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} else {
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"Failed to reach target accuracy"
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}
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));
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console::log_2(&JsValue::from("Steps:"), &JsValue::from(history.scaled_loss.len() - 1));
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console::log_2(&JsValue::from("Loss:"), &JsValue::from(*history.scaled_loss.last().unwrap()));
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if success {
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// read out the solution
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self.elements.update(|elts| {
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for (_, elt) in elts.iter_mut() {
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elt.rep.set_column(0, &config.column(elt.index));
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}
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});
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}
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}
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}
|
@ -1,12 +1,4 @@
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use lazy_static::lazy_static;
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use nalgebra::{Const, DMatrix, DVector, Dyn};
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use web_sys::{console, wasm_bindgen::JsValue}; /* DEBUG */
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// --- elements ---
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pub fn point(x: f64, y: f64, z: f64) -> DVector<f64> {
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DVector::from_column_slice(&[x, y, z, 0.5, 0.5*(x*x + y*y + z*z)])
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}
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use nalgebra::DVector;
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// the sphere with the given center and radius, with inward-pointing normals
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pub fn sphere(center_x: f64, center_y: f64, center_z: f64, radius: f64) -> DVector<f64> {
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@ -33,470 +25,3 @@ pub fn sphere_with_offset(dir_x: f64, dir_y: f64, dir_z: f64, off: f64, curv: f6
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off * (1.0 + 0.5 * off * curv)
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])
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}
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// --- partial matrices ---
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struct MatrixEntry {
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index: (usize, usize),
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value: f64
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}
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pub struct PartialMatrix(Vec<MatrixEntry>);
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impl PartialMatrix {
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pub fn new() -> PartialMatrix {
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PartialMatrix(Vec::<MatrixEntry>::new())
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}
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pub fn push_sym(&mut self, row: usize, col: usize, value: f64) {
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let PartialMatrix(entries) = self;
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entries.push(MatrixEntry { index: (row, col), value: value });
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if row != col {
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entries.push(MatrixEntry { index: (col, row), value: value });
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}
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}
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/* DEBUG */
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pub fn log_to_console(&self) {
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let PartialMatrix(entries) = self;
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for ent in entries {
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let ent_str = format!("{} {} {}", ent.index.0, ent.index.1, ent.value);
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console::log_1(&JsValue::from(ent_str.as_str()));
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}
|
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}
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fn proj(&self, a: &DMatrix<f64>) -> DMatrix<f64> {
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let mut result = DMatrix::<f64>::zeros(a.nrows(), a.ncols());
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let PartialMatrix(entries) = self;
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for ent in entries {
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result[ent.index] = a[ent.index];
|
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}
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result
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}
|
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|
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fn sub_proj(&self, rhs: &DMatrix<f64>) -> DMatrix<f64> {
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let mut result = DMatrix::<f64>::zeros(rhs.nrows(), rhs.ncols());
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let PartialMatrix(entries) = self;
|
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for ent in entries {
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result[ent.index] = ent.value - rhs[ent.index];
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}
|
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result
|
||||
}
|
||||
}
|
||||
|
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// --- descent history ---
|
||||
|
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pub struct DescentHistory {
|
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pub config: Vec<DMatrix<f64>>,
|
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pub scaled_loss: Vec<f64>,
|
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pub neg_grad: Vec<DMatrix<f64>>,
|
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pub min_eigval: Vec<f64>,
|
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pub base_step: Vec<DMatrix<f64>>,
|
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pub backoff_steps: Vec<i32>
|
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}
|
||||
|
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impl DescentHistory {
|
||||
fn new() -> DescentHistory {
|
||||
DescentHistory {
|
||||
config: Vec::<DMatrix<f64>>::new(),
|
||||
scaled_loss: Vec::<f64>::new(),
|
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neg_grad: Vec::<DMatrix<f64>>::new(),
|
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min_eigval: Vec::<f64>::new(),
|
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base_step: Vec::<DMatrix<f64>>::new(),
|
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backoff_steps: Vec::<i32>::new(),
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
// --- gram matrix realization ---
|
||||
|
||||
// the Lorentz form
|
||||
lazy_static! {
|
||||
static ref Q: DMatrix<f64> = DMatrix::from_row_slice(5, 5, &[
|
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1.0, 0.0, 0.0, 0.0, 0.0,
|
||||
0.0, 1.0, 0.0, 0.0, 0.0,
|
||||
0.0, 0.0, 1.0, 0.0, 0.0,
|
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0.0, 0.0, 0.0, 0.0, -2.0,
|
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0.0, 0.0, 0.0, -2.0, 0.0
|
||||
]);
|
||||
}
|
||||
|
||||
struct SearchState {
|
||||
config: DMatrix<f64>,
|
||||
err_proj: DMatrix<f64>,
|
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loss: f64
|
||||
}
|
||||
|
||||
impl SearchState {
|
||||
fn from_config(gram: &PartialMatrix, config: DMatrix<f64>) -> SearchState {
|
||||
let err_proj = gram.sub_proj(&(config.tr_mul(&*Q) * &config));
|
||||
let loss = err_proj.norm_squared();
|
||||
SearchState {
|
||||
config: config,
|
||||
err_proj: err_proj,
|
||||
loss: loss
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
fn basis_matrix(index: (usize, usize), nrows: usize, ncols: usize) -> DMatrix<f64> {
|
||||
let mut result = DMatrix::<f64>::zeros(nrows, ncols);
|
||||
result[index] = 1.0;
|
||||
result
|
||||
}
|
||||
|
||||
// use backtracking line search to find a better configuration
|
||||
fn seek_better_config(
|
||||
gram: &PartialMatrix,
|
||||
state: &SearchState,
|
||||
base_step: &DMatrix<f64>,
|
||||
base_target_improvement: f64,
|
||||
min_efficiency: f64,
|
||||
backoff: f64,
|
||||
max_backoff_steps: i32
|
||||
) -> Option<(SearchState, i32)> {
|
||||
let mut rate = 1.0;
|
||||
for backoff_steps in 0..max_backoff_steps {
|
||||
let trial_config = &state.config + rate * base_step;
|
||||
let trial_state = SearchState::from_config(gram, trial_config);
|
||||
let improvement = state.loss - trial_state.loss;
|
||||
if improvement >= min_efficiency * rate * base_target_improvement {
|
||||
return Some((trial_state, backoff_steps));
|
||||
}
|
||||
rate *= backoff;
|
||||
}
|
||||
None
|
||||
}
|
||||
|
||||
// seek a matrix `config` for which `config' * Q * config` matches the partial
|
||||
// matrix `gram`. use gradient descent starting from `guess`
|
||||
pub fn realize_gram(
|
||||
gram: &PartialMatrix,
|
||||
guess: DMatrix<f64>,
|
||||
frozen: &[(usize, usize)],
|
||||
scaled_tol: f64,
|
||||
min_efficiency: f64,
|
||||
backoff: f64,
|
||||
reg_scale: f64,
|
||||
max_descent_steps: i32,
|
||||
max_backoff_steps: i32
|
||||
) -> (DMatrix<f64>, bool, DescentHistory) {
|
||||
// start the descent history
|
||||
let mut history = DescentHistory::new();
|
||||
|
||||
// find the dimension of the search space
|
||||
let element_dim = guess.nrows();
|
||||
let assembly_dim = guess.ncols();
|
||||
let total_dim = element_dim * assembly_dim;
|
||||
|
||||
// scale the tolerance
|
||||
let scale_adjustment = (gram.0.len() as f64).sqrt();
|
||||
let tol = scale_adjustment * scaled_tol;
|
||||
|
||||
// convert the frozen indices to stacked format
|
||||
let frozen_stacked: Vec<usize> = frozen.into_iter().map(
|
||||
|index| index.1*element_dim + index.0
|
||||
).collect();
|
||||
|
||||
// use Newton's method with backtracking and gradient descent backup
|
||||
let mut state = SearchState::from_config(gram, guess);
|
||||
for _ in 0..max_descent_steps {
|
||||
// stop if the loss is tolerably low
|
||||
history.config.push(state.config.clone());
|
||||
history.scaled_loss.push(state.loss / scale_adjustment);
|
||||
if state.loss < tol { break; }
|
||||
|
||||
// find the negative gradient of the loss function
|
||||
let neg_grad = 4.0 * &*Q * &state.config * &state.err_proj;
|
||||
let mut neg_grad_stacked = neg_grad.clone().reshape_generic(Dyn(total_dim), Const::<1>);
|
||||
history.neg_grad.push(neg_grad.clone());
|
||||
|
||||
// find the negative Hessian of the loss function
|
||||
let mut hess_cols = Vec::<DVector<f64>>::with_capacity(total_dim);
|
||||
for col in 0..assembly_dim {
|
||||
for row in 0..element_dim {
|
||||
let index = (row, col);
|
||||
let basis_mat = basis_matrix(index, element_dim, assembly_dim);
|
||||
let neg_d_err =
|
||||
basis_mat.tr_mul(&*Q) * &state.config
|
||||
+ state.config.tr_mul(&*Q) * &basis_mat;
|
||||
let neg_d_err_proj = gram.proj(&neg_d_err);
|
||||
let deriv_grad = 4.0 * &*Q * (
|
||||
-&basis_mat * &state.err_proj
|
||||
+ &state.config * &neg_d_err_proj
|
||||
);
|
||||
hess_cols.push(deriv_grad.reshape_generic(Dyn(total_dim), Const::<1>));
|
||||
}
|
||||
}
|
||||
let mut hess = DMatrix::from_columns(hess_cols.as_slice());
|
||||
|
||||
// regularize the Hessian
|
||||
let min_eigval = hess.symmetric_eigenvalues().min();
|
||||
if min_eigval <= 0.0 {
|
||||
hess -= reg_scale * min_eigval * DMatrix::identity(total_dim, total_dim);
|
||||
}
|
||||
history.min_eigval.push(min_eigval);
|
||||
|
||||
// project the negative gradient and negative Hessian onto the
|
||||
// orthogonal complement of the frozen subspace
|
||||
let zero_col = DVector::zeros(total_dim);
|
||||
let zero_row = zero_col.transpose();
|
||||
for &k in &frozen_stacked {
|
||||
neg_grad_stacked[k] = 0.0;
|
||||
hess.set_row(k, &zero_row);
|
||||
hess.set_column(k, &zero_col);
|
||||
hess[(k, k)] = 1.0;
|
||||
}
|
||||
|
||||
// compute the Newton step
|
||||
/*
|
||||
we need to either handle or eliminate the case where the minimum
|
||||
eigenvalue of the Hessian is zero, so the regularized Hessian is
|
||||
singular. right now, this causes the Cholesky decomposition to return
|
||||
`None`, leading to a panic when we unrap
|
||||
*/
|
||||
let base_step_stacked = hess.cholesky().unwrap().solve(&neg_grad_stacked);
|
||||
let base_step = base_step_stacked.reshape_generic(Dyn(element_dim), Dyn(assembly_dim));
|
||||
history.base_step.push(base_step.clone());
|
||||
|
||||
// use backtracking line search to find a better configuration
|
||||
match seek_better_config(
|
||||
gram, &state, &base_step, neg_grad.dot(&base_step),
|
||||
min_efficiency, backoff, max_backoff_steps
|
||||
) {
|
||||
Some((better_state, backoff_steps)) => {
|
||||
state = better_state;
|
||||
history.backoff_steps.push(backoff_steps);
|
||||
},
|
||||
None => return (state.config, false, history)
|
||||
};
|
||||
}
|
||||
(state.config, state.loss < tol, history)
|
||||
}
|
||||
|
||||
// --- tests ---
|
||||
|
||||
#[cfg(test)]
|
||||
mod tests {
|
||||
use std::{array, f64::consts::PI};
|
||||
|
||||
use super::*;
|
||||
|
||||
#[test]
|
||||
fn sub_proj_test() {
|
||||
let target = PartialMatrix(vec![
|
||||
MatrixEntry { index: (0, 0), value: 19.0 },
|
||||
MatrixEntry { index: (0, 2), value: 39.0 },
|
||||
MatrixEntry { index: (1, 1), value: 59.0 },
|
||||
MatrixEntry { index: (1, 2), value: 69.0 }
|
||||
]);
|
||||
let attempt = DMatrix::<f64>::from_row_slice(2, 3, &[
|
||||
1.0, 2.0, 3.0,
|
||||
4.0, 5.0, 6.0
|
||||
]);
|
||||
let expected_result = DMatrix::<f64>::from_row_slice(2, 3, &[
|
||||
18.0, 0.0, 36.0,
|
||||
0.0, 54.0, 63.0
|
||||
]);
|
||||
assert_eq!(target.sub_proj(&attempt), expected_result);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn zero_loss_test() {
|
||||
let gram = PartialMatrix({
|
||||
let mut entries = Vec::<MatrixEntry>::new();
|
||||
for j in 0..3 {
|
||||
for k in 0..3 {
|
||||
entries.push(MatrixEntry {
|
||||
index: (j, k),
|
||||
value: if j == k { 1.0 } else { -1.0 }
|
||||
});
|
||||
}
|
||||
}
|
||||
entries
|
||||
});
|
||||
let config = {
|
||||
let a: f64 = 0.75_f64.sqrt();
|
||||
DMatrix::from_columns(&[
|
||||
sphere(1.0, 0.0, 0.0, a),
|
||||
sphere(-0.5, a, 0.0, a),
|
||||
sphere(-0.5, -a, 0.0, a)
|
||||
])
|
||||
};
|
||||
let state = SearchState::from_config(&gram, config);
|
||||
assert!(state.loss.abs() < f64::EPSILON);
|
||||
}
|
||||
|
||||
// this problem is from a sangaku by Irisawa Shintarō Hiroatsu. the article
|
||||
// below includes a nice translation of the problem statement, which was
|
||||
// recorded in Uchida Itsumi's book _Kokon sankan_ (_Mathematics, Past and
|
||||
// Present_)
|
||||
//
|
||||
// "Japan's 'Wasan' Mathematical Tradition", by Abe Haruki
|
||||
// https://www.nippon.com/en/japan-topics/c12801/
|
||||
//
|
||||
#[test]
|
||||
fn irisawa_hexlet_test() {
|
||||
let gram = PartialMatrix({
|
||||
let mut entries = Vec::<MatrixEntry>::new();
|
||||
for s in 0..9 {
|
||||
// each sphere is represented by a spacelike vector
|
||||
entries.push(MatrixEntry { index: (s, s), value: 1.0 });
|
||||
|
||||
// the circumscribing sphere is tangent to all of the other
|
||||
// spheres, with matching orientation
|
||||
if s > 0 {
|
||||
entries.push(MatrixEntry { index: (0, s), value: 1.0 });
|
||||
entries.push(MatrixEntry { index: (s, 0), value: 1.0 });
|
||||
}
|
||||
|
||||
if s > 2 {
|
||||
// each chain sphere is tangent to the "sun" and "moon"
|
||||
// spheres, with opposing orientation
|
||||
for n in 1..3 {
|
||||
entries.push(MatrixEntry { index: (s, n), value: -1.0 });
|
||||
entries.push(MatrixEntry { index: (n, s), value: -1.0 });
|
||||
}
|
||||
|
||||
// each chain sphere is tangent to the next chain sphere,
|
||||
// with opposing orientation
|
||||
let s_next = 3 + (s-2) % 6;
|
||||
entries.push(MatrixEntry { index: (s, s_next), value: -1.0 });
|
||||
entries.push(MatrixEntry { index: (s_next, s), value: -1.0 });
|
||||
}
|
||||
}
|
||||
entries
|
||||
});
|
||||
let guess = DMatrix::from_columns(
|
||||
[
|
||||
sphere(0.0, 0.0, 0.0, 15.0),
|
||||
sphere(0.0, 0.0, -9.0, 5.0),
|
||||
sphere(0.0, 0.0, 11.0, 3.0)
|
||||
].into_iter().chain(
|
||||
(1..=6).map(
|
||||
|k| {
|
||||
let ang = (k as f64) * PI/3.0;
|
||||
sphere(9.0 * ang.cos(), 9.0 * ang.sin(), 0.0, 2.5)
|
||||
}
|
||||
)
|
||||
).collect::<Vec<_>>().as_slice()
|
||||
);
|
||||
let frozen: [(usize, usize); 4] = array::from_fn(|k| (3, k));
|
||||
const SCALED_TOL: f64 = 1.0e-12;
|
||||
let (config, success, history) = realize_gram(
|
||||
&gram, guess, &frozen,
|
||||
SCALED_TOL, 0.5, 0.9, 1.1, 200, 110
|
||||
);
|
||||
let entry_tol = SCALED_TOL.sqrt();
|
||||
let solution_diams = [30.0, 10.0, 6.0, 5.0, 15.0, 10.0, 3.75, 2.5, 2.0 + 8.0/11.0];
|
||||
for (k, diam) in solution_diams.into_iter().enumerate() {
|
||||
assert!((config[(3, k)] - 1.0 / diam).abs() < entry_tol);
|
||||
}
|
||||
print!("\nCompleted Gram matrix:{}", config.tr_mul(&*Q) * &config);
|
||||
if success {
|
||||
println!("Target accuracy achieved!");
|
||||
} else {
|
||||
println!("Failed to reach target accuracy");
|
||||
}
|
||||
println!("Steps: {}", history.scaled_loss.len() - 1);
|
||||
println!("Loss: {}", history.scaled_loss.last().unwrap());
|
||||
if success {
|
||||
println!("\nChain diameters:");
|
||||
println!(" {} sun (given)", 1.0 / config[(3, 3)]);
|
||||
for k in 4..9 {
|
||||
println!(" {} sun", 1.0 / config[(3, k)]);
|
||||
}
|
||||
}
|
||||
println!("\nStep │ Loss\n─────┼────────────────────────────────");
|
||||
for (step, scaled_loss) in history.scaled_loss.into_iter().enumerate() {
|
||||
println!("{:<4} │ {}", step, scaled_loss);
|
||||
}
|
||||
}
|
||||
|
||||
// --- process inspection examples ---
|
||||
|
||||
// these tests are meant for human inspection, not automated use. run them
|
||||
// one at a time in `--nocapture` mode and read through the results and
|
||||
// optimization histories that they print out. the `run-examples` script
|
||||
// will run all of them
|
||||
|
||||
#[test]
|
||||
fn three_spheres_example() {
|
||||
let gram = PartialMatrix({
|
||||
let mut entries = Vec::<MatrixEntry>::new();
|
||||
for j in 0..3 {
|
||||
for k in 0..3 {
|
||||
entries.push(MatrixEntry {
|
||||
index: (j, k),
|
||||
value: if j == k { 1.0 } else { -1.0 }
|
||||
});
|
||||
}
|
||||
}
|
||||
entries
|
||||
});
|
||||
let guess = {
|
||||
let a: f64 = 0.75_f64.sqrt();
|
||||
DMatrix::from_columns(&[
|
||||
sphere(1.0, 0.0, 0.0, 1.0),
|
||||
sphere(-0.5, a, 0.0, 1.0),
|
||||
sphere(-0.5, -a, 0.0, 1.0)
|
||||
])
|
||||
};
|
||||
println!();
|
||||
let (config, success, history) = realize_gram(
|
||||
&gram, guess, &[],
|
||||
1.0e-12, 0.5, 0.9, 1.1, 200, 110
|
||||
);
|
||||
print!("\nCompleted Gram matrix:{}", config.tr_mul(&*Q) * &config);
|
||||
if success {
|
||||
println!("Target accuracy achieved!");
|
||||
} else {
|
||||
println!("Failed to reach target accuracy");
|
||||
}
|
||||
println!("Steps: {}", history.scaled_loss.len() - 1);
|
||||
println!("Loss: {}", history.scaled_loss.last().unwrap());
|
||||
println!("\nStep │ Loss\n─────┼────────────────────────────────");
|
||||
for (step, scaled_loss) in history.scaled_loss.into_iter().enumerate() {
|
||||
println!("{:<4} │ {}", step, scaled_loss);
|
||||
}
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn point_on_sphere_example() {
|
||||
let gram = PartialMatrix({
|
||||
let mut entries = Vec::<MatrixEntry>::new();
|
||||
for j in 0..2 {
|
||||
for k in 0..2 {
|
||||
entries.push(MatrixEntry {
|
||||
index: (j, k),
|
||||
value: if (j, k) == (1, 1) { 1.0 } else { 0.0 }
|
||||
});
|
||||
}
|
||||
}
|
||||
entries
|
||||
});
|
||||
let guess = DMatrix::from_columns(&[
|
||||
point(0.0, 0.0, 2.0),
|
||||
sphere(0.0, 0.0, 0.0, 1.0)
|
||||
]);
|
||||
let frozen = [(3, 0)];
|
||||
println!();
|
||||
let (config, success, history) = realize_gram(
|
||||
&gram, guess, &frozen,
|
||||
1.0e-12, 0.5, 0.9, 1.1, 200, 110
|
||||
);
|
||||
print!("\nCompleted Gram matrix:{}", config.tr_mul(&*Q) * &config);
|
||||
print!("Configuration:{}", config);
|
||||
if success {
|
||||
println!("Target accuracy achieved!");
|
||||
} else {
|
||||
println!("Failed to reach target accuracy");
|
||||
}
|
||||
println!("Steps: {}", history.scaled_loss.len() - 1);
|
||||
println!("Loss: {}", history.scaled_loss.last().unwrap());
|
||||
println!("\nStep │ Loss\n─────┼────────────────────────────────");
|
||||
for (step, scaled_loss) in history.scaled_loss.into_iter().enumerate() {
|
||||
println!("{:<4} │ {}", step, scaled_loss);
|
||||
}
|
||||
}
|
||||
}
|
@ -8,8 +8,7 @@ using Optim
|
||||
|
||||
export
|
||||
rand_on_shell, Q, DescentHistory,
|
||||
realize_gram_gradient, realize_gram_newton, realize_gram_optim,
|
||||
realize_gram_alt_proj, realize_gram
|
||||
realize_gram_gradient, realize_gram_newton, realize_gram_optim, realize_gram
|
||||
|
||||
# === guessing ===
|
||||
|
||||
@ -144,7 +143,7 @@ function realize_gram_gradient(
|
||||
break
|
||||
end
|
||||
|
||||
# find the negative gradient of the loss function
|
||||
# find negative gradient of loss function
|
||||
neg_grad = 4*Q*L*Δ_proj
|
||||
slope = norm(neg_grad)
|
||||
dir = neg_grad / slope
|
||||
@ -233,7 +232,7 @@ function realize_gram_newton(
|
||||
break
|
||||
end
|
||||
|
||||
# find the negative gradient of the loss function
|
||||
# find the negative gradient of loss function
|
||||
neg_grad = 4*Q*L*Δ_proj
|
||||
|
||||
# find the negative Hessian of the loss function
|
||||
@ -314,129 +313,6 @@ function realize_gram_optim(
|
||||
)
|
||||
end
|
||||
|
||||
# seek a matrix `L` for which `L'QL` matches the sparse matrix `gram` at every
|
||||
# explicit entry of `gram`. use gradient descent starting from `guess`, with an
|
||||
# alternate technique for finding the projected base step from the unprojected
|
||||
# Hessian
|
||||
function realize_gram_alt_proj(
|
||||
gram::SparseMatrixCSC{T, <:Any},
|
||||
guess::Matrix{T},
|
||||
frozen = CartesianIndex[];
|
||||
scaled_tol = 1e-30,
|
||||
min_efficiency = 0.5,
|
||||
backoff = 0.9,
|
||||
reg_scale = 1.1,
|
||||
max_descent_steps = 200,
|
||||
max_backoff_steps = 110
|
||||
) where T <: Number
|
||||
# start history
|
||||
history = DescentHistory{T}()
|
||||
|
||||
# find the dimension of the search space
|
||||
dims = size(guess)
|
||||
element_dim, construction_dim = dims
|
||||
total_dim = element_dim * construction_dim
|
||||
|
||||
# list the constrained entries of the gram matrix
|
||||
J, K, _ = findnz(gram)
|
||||
constrained = zip(J, K)
|
||||
|
||||
# scale the tolerance
|
||||
scale_adjustment = sqrt(T(length(constrained)))
|
||||
tol = scale_adjustment * scaled_tol
|
||||
|
||||
# convert the frozen indices to stacked format
|
||||
frozen_stacked = [(index[2]-1)*element_dim + index[1] for index in frozen]
|
||||
|
||||
# initialize search state
|
||||
L = copy(guess)
|
||||
Δ_proj = proj_diff(gram, L'*Q*L)
|
||||
loss = dot(Δ_proj, Δ_proj)
|
||||
|
||||
# use Newton's method with backtracking and gradient descent backup
|
||||
for step in 1:max_descent_steps
|
||||
# stop if the loss is tolerably low
|
||||
if loss < tol
|
||||
break
|
||||
end
|
||||
|
||||
# find the negative gradient of the loss function
|
||||
neg_grad = 4*Q*L*Δ_proj
|
||||
|
||||
# find the negative Hessian of the loss function
|
||||
hess = Matrix{T}(undef, total_dim, total_dim)
|
||||
indices = [(j, k) for k in 1:construction_dim for j in 1:element_dim]
|
||||
for (j, k) in indices
|
||||
basis_mat = basis_matrix(T, j, k, dims)
|
||||
neg_dΔ = basis_mat'*Q*L + L'*Q*basis_mat
|
||||
neg_dΔ_proj = proj_to_entries(neg_dΔ, constrained)
|
||||
deriv_grad = 4*Q*(-basis_mat*Δ_proj + L*neg_dΔ_proj)
|
||||
hess[:, (k-1)*element_dim + j] = reshape(deriv_grad, total_dim)
|
||||
end
|
||||
hess_sym = Hermitian(hess)
|
||||
push!(history.hess, hess_sym)
|
||||
|
||||
# regularize the Hessian
|
||||
min_eigval = minimum(eigvals(hess_sym))
|
||||
push!(history.positive, min_eigval > 0)
|
||||
if min_eigval <= 0
|
||||
hess -= reg_scale * min_eigval * I
|
||||
end
|
||||
|
||||
# compute the Newton step
|
||||
neg_grad_stacked = reshape(neg_grad, total_dim)
|
||||
for k in frozen_stacked
|
||||
neg_grad_stacked[k] = 0
|
||||
hess[k, :] .= 0
|
||||
hess[:, k] .= 0
|
||||
hess[k, k] = 1
|
||||
end
|
||||
base_step_stacked = Hermitian(hess) \ neg_grad_stacked
|
||||
base_step = reshape(base_step_stacked, dims)
|
||||
push!(history.base_step, base_step)
|
||||
|
||||
# store the current position, loss, and slope
|
||||
L_last = L
|
||||
loss_last = loss
|
||||
push!(history.scaled_loss, loss / scale_adjustment)
|
||||
push!(history.neg_grad, neg_grad)
|
||||
push!(history.slope, norm(neg_grad))
|
||||
|
||||
# find a good step size using backtracking line search
|
||||
push!(history.stepsize, 0)
|
||||
push!(history.backoff_steps, max_backoff_steps)
|
||||
empty!(history.last_line_L)
|
||||
empty!(history.last_line_loss)
|
||||
rate = one(T)
|
||||
step_success = false
|
||||
base_target_improvement = dot(neg_grad, base_step)
|
||||
for backoff_steps in 0:max_backoff_steps
|
||||
history.stepsize[end] = rate
|
||||
L = L_last + rate * base_step
|
||||
Δ_proj = proj_diff(gram, L'*Q*L)
|
||||
loss = dot(Δ_proj, Δ_proj)
|
||||
improvement = loss_last - loss
|
||||
push!(history.last_line_L, L)
|
||||
push!(history.last_line_loss, loss / scale_adjustment)
|
||||
if improvement >= min_efficiency * rate * base_target_improvement
|
||||
history.backoff_steps[end] = backoff_steps
|
||||
step_success = true
|
||||
break
|
||||
end
|
||||
rate *= backoff
|
||||
end
|
||||
|
||||
# if we've hit a wall, quit
|
||||
if !step_success
|
||||
return L_last, false, history
|
||||
end
|
||||
end
|
||||
|
||||
# return the factorization and its history
|
||||
push!(history.scaled_loss, loss / scale_adjustment)
|
||||
L, loss < tol, history
|
||||
end
|
||||
|
||||
# seek a matrix `L` for which `L'QL` matches the sparse matrix `gram` at every
|
||||
# explicit entry of `gram`. use gradient descent starting from `guess`
|
||||
function realize_gram(
|
||||
@ -445,6 +321,7 @@ function realize_gram(
|
||||
frozen = nothing;
|
||||
scaled_tol = 1e-30,
|
||||
min_efficiency = 0.5,
|
||||
init_rate = 1.0,
|
||||
backoff = 0.9,
|
||||
reg_scale = 1.1,
|
||||
max_descent_steps = 200,
|
||||
@ -475,19 +352,20 @@ function realize_gram(
|
||||
unfrozen_stacked = reshape(is_unfrozen, total_dim)
|
||||
end
|
||||
|
||||
# initialize search state
|
||||
# initialize variables
|
||||
grad_rate = init_rate
|
||||
L = copy(guess)
|
||||
Δ_proj = proj_diff(gram, L'*Q*L)
|
||||
loss = dot(Δ_proj, Δ_proj)
|
||||
|
||||
# use Newton's method with backtracking and gradient descent backup
|
||||
Δ_proj = proj_diff(gram, L'*Q*L)
|
||||
loss = dot(Δ_proj, Δ_proj)
|
||||
for step in 1:max_descent_steps
|
||||
# stop if the loss is tolerably low
|
||||
if loss < tol
|
||||
break
|
||||
end
|
||||
|
||||
# find the negative gradient of the loss function
|
||||
# find the negative gradient of loss function
|
||||
neg_grad = 4*Q*L*Δ_proj
|
||||
|
||||
# find the negative Hessian of the loss function
|
||||
@ -542,7 +420,6 @@ function realize_gram(
|
||||
empty!(history.last_line_loss)
|
||||
rate = one(T)
|
||||
step_success = false
|
||||
base_target_improvement = dot(neg_grad, base_step)
|
||||
for backoff_steps in 0:max_backoff_steps
|
||||
history.stepsize[end] = rate
|
||||
L = L_last + rate * base_step
|
||||
@ -551,7 +428,7 @@ function realize_gram(
|
||||
improvement = loss_last - loss
|
||||
push!(history.last_line_L, L)
|
||||
push!(history.last_line_loss, loss / scale_adjustment)
|
||||
if improvement >= min_efficiency * rate * base_target_improvement
|
||||
if improvement >= min_efficiency * rate * dot(neg_grad, base_step)
|
||||
history.backoff_steps[end] = backoff_steps
|
||||
step_success = true
|
||||
break
|
||||
|
@ -75,12 +75,3 @@ if success
|
||||
println(" ", 1 / L[4,k], " sun")
|
||||
end
|
||||
end
|
||||
|
||||
# test an alternate technique for finding the projected base step from the
|
||||
# unprojected Hessian
|
||||
L_alt, success_alt, history_alt = Engine.realize_gram_alt_proj(gram, guess, frozen)
|
||||
completed_gram_alt = L_alt'*Engine.Q*L_alt
|
||||
println("\nDifference in result using alternate projection:\n")
|
||||
display(completed_gram_alt - completed_gram)
|
||||
println("\nDifference in steps: ", size(history_alt.scaled_loss, 1) - size(history.scaled_loss, 1))
|
||||
println("Difference in loss: ", history_alt.scaled_loss[end] - history.scaled_loss[end], "\n")
|
@ -65,12 +65,3 @@ else
|
||||
end
|
||||
println("Steps: ", size(history.scaled_loss, 1))
|
||||
println("Loss: ", history.scaled_loss[end], "\n")
|
||||
|
||||
# test an alternate technique for finding the projected base step from the
|
||||
# unprojected Hessian
|
||||
L_alt, success_alt, history_alt = Engine.realize_gram_alt_proj(gram, guess, frozen)
|
||||
completed_gram_alt = L_alt'*Engine.Q*L_alt
|
||||
println("\nDifference in result using alternate projection:\n")
|
||||
display(completed_gram_alt - completed_gram)
|
||||
println("\nDifference in steps: ", size(history_alt.scaled_loss, 1) - size(history.scaled_loss, 1))
|
||||
println("Difference in loss: ", history_alt.scaled_loss[end] - history.scaled_loss[end], "\n")
|
@ -94,12 +94,3 @@ if success
|
||||
radius_ratio = dot(infty, Engine.Q * L[:,5]) / dot(infty, Engine.Q * L[:,6])
|
||||
println("\nCircumradius / inradius: ", radius_ratio)
|
||||
end
|
||||
|
||||
# test an alternate technique for finding the projected base step from the
|
||||
# unprojected Hessian
|
||||
L_alt, success_alt, history_alt = Engine.realize_gram_alt_proj(gram, guess, frozen)
|
||||
completed_gram_alt = L_alt'*Engine.Q*L_alt
|
||||
println("\nDifference in result using alternate projection:\n")
|
||||
display(completed_gram_alt - completed_gram)
|
||||
println("\nDifference in steps: ", size(history_alt.scaled_loss, 1) - size(history.scaled_loss, 1))
|
||||
println("Difference in loss: ", history_alt.scaled_loss[end] - history.scaled_loss[end], "\n")
|
Loading…
Reference in New Issue
Block a user