RefineHartiganWong.hpp

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#ifndef KMEANS_HARTIGAN_WONG_HPP
#define KMEANS_HARTIGAN_WONG_HPP
 
#include <vector>
#include <algorithm>
#include <numeric>
#include <cstdint>
 
#include "Refine.hpp"
#include "Details.hpp"
#include "QuickSearch.hpp"
#include "parallelize.hpp"
#include "compute_centroids.hpp"
#include "is_edge_case.hpp"
 
namespace kmeans {
 
struct RefineHartiganWongOptions {
    int max_iterations = 10;
 
    int max_quick_transfer_iterations = 50;
 
    bool quit_on_quick_transfer_convergence_failure = false;
 
    int num_threads = 1;
};
 
namespace RefineHartiganWong_internal {
 
/*
 * Alright, get ready for a data dump, because this is where it gets real.
 * Here I attempt to untangle the code spaghetti from the original Fortran
 * implementation, which is exemplified by the NCP and LIVE arrays.
 *
 * Okay, NCP first. Each element L in NCP has a dual interpretation:
 *
 * - In the optimal-transfer stage, NCP(L) stores the step at which cluster L
 *   was last updated. Each step is just the observation index as the optimal
 *   transfer only does one pass through the dataset.
 * - In the quick-transfer stage, NCP(L) stores the step at which cluster L is
 *   last updated plus M (i.e., the number of observations). Here, the step
 *   corresponding to an observation will be 'M * X + obs' for some integer X
 *   >= 0, where X is the iteration of the quick transfer. This means that the
 *   stored value is defined as '(M + 1) * X + obs'.
 * - After each quick_transfer call, NCP(L) is set back to zero so any existing
 *   values will not carry over into the next optimal_transfer call.
 *
 * Note that these two definitions bleed into each other as the NCP(L) set by
 * optimal_transfer is still referenced in the first few iterations of
 * quick_transfer before it eventually gets overwritten. The easiest way to
 * interpret this is to consider the optimal transfer as iteration 'M = -1'
 * from the perspective of the quick transfer iterations. 
 *
 * In short, the NCP array is used to determine whether a cluster was modified
 * within the last M steps of the optimal_transfer or quick_transfer.
 *
 * Let's move onto LIVE, which has an even more painful interpretation:
 *
 * - Before each optimal_transfer call, LIVE(L) stores the observation at which
 *   cluster L was updated in the _previous_ optimal_transfer call.
 * - During the optimal_transfer call, LIVE(L) is updated to the observation at
 *   which L was updated in this call, plus M (i.e., number of observations).
 * - After the optimal_transfer call, LIVE(L) is updated by subtracting M, so
 *   that the interpretation is correct in the next call.
 * - After the quick_transfer call, LIVE(L) is set to M + 1 if a quick transfer
 *   took place for L, effectively mimicking a "very recent" update.
 *
 * It basically tells us whether there was a recent transfer (optimal or quick)
 * within the last M steps of optimal_transfer. If so, the cluster is "live".
 * This differs very slightly from NCP as LIVE only counts optimal transfer
 * steps while NCP counts both optimal and quick transfer steps.
 *
 * We simplify this mess by consolidating NCP and LIVE into a single update
 * history that can serve both purposes. This is possible as LIVE is not used
 * in quick_transfer, while any modifications made to NCP by quick_transfer are
 * ignored and overwritten in optimal_transfer. Thus, there won't be any
 * conflicts between the two meanings in a consolidated update history.
 *
 * The update history for cluster L now holds the iteration ('X') and the
 * observation ('obs') at which the cluster was last modified. This has
 * an obvious mapping to NCP(L), which was already defined in terms of 
 * 'X' and 'obs' anyway. For LIVE(L), the mapping is more complex:
 *
 * - If an optimal_transfer occurred but no quick_transfer occurs for L,
 *   we set 'X = -2'. 'obs' is not modified as it is the observation at
 *   which L was modified in the previous optimal_transfer iteration;
 *   this is basically just the definition of LIVE(L).
 * - If a quick transfer does occur, we set 'X = -2' and 'obs = M + 1'.
 *   Again, this is equivalent to the behavior of LIVE(L).
 * - If neither a quick_transfer nor optimal_transfer occurs for L,
 *   we set 'X = -3' to indicate that L didn't change in any manner.
 * - During the optimal_transfer, we consider a cluster to be live at a
 *   particular observation if it is greater than 'obs'.
 *
 * Note that Fortran is 1-based so the actual code is a little different than
 * described above for 'obs' - specifically, we just need to set it to 'M'
 * when a quick transfer occurs. Meaning of 'X' is unchanged.
 *
 * Incidentally, we can easily detect if a quick transfer occurs based on
 * whether 'X > -1'. This obviates the need for the ITRAN array.
 */
template<typename Index_>
class UpdateHistory {
private:
    Index_ my_last_observation = 0;
 
    static constexpr int current_optimal_transfer = -1;
    static constexpr int previous_optimal_transfer = -2;
    static constexpr int ancient_history = -3;
 
    // Starting at -3 as we can't possibly have any updates if we haven't done
    // any transfers, optimal or quick!
    int my_last_iteration = ancient_history;
 
public:
    void reset(Index_ total_obs) {
        if (my_last_iteration > current_optimal_transfer) { // i.e., quick_transfer.
            my_last_observation = total_obs;
            my_last_iteration = previous_optimal_transfer;
        } else if (my_last_iteration == current_optimal_transfer) {
            // Preserve the existing 'my_last_observation', just bump the iteration downwards.
            my_last_iteration = previous_optimal_transfer;
        } else {
            my_last_iteration = ancient_history;
        }
    }
 
    void set_optimal(Index_ obs) {
        my_last_observation = obs;
        my_last_iteration = current_optimal_transfer;
    }
 
    // Here, iter should be from '[0, max_quick_transfer_iterations)'.
    void set_quick(int iter, Index_ obs) {
        my_last_iteration = iter;
        my_last_observation = obs;
    }
 
public:
    bool changed_after(int iter, Index_ obs) const {
        if (my_last_iteration == iter) {
            return my_last_observation > obs;
        } else {
            return my_last_iteration > iter;
        }
    }
 
    bool changed_after_or_at(int iter, Index_ obs) const {
        if (my_last_iteration == iter) {
            return my_last_observation >= obs;
        } else {
            return my_last_iteration > iter;
        }
    }
 
    bool is_live(Index_ obs) const {
        return changed_after(previous_optimal_transfer, obs);
    }
};
 
template<typename Float_, typename Index_, typename Cluster_>
struct Workspace {
    // Array arguments in the same order as supplied to R's kmns_ function.
    std::vector<Cluster_> best_destination_cluster; // i.e., IC2
    std::vector<Index_> cluster_sizes; // i.e., NC
 
    std::vector<Float_> loss_multiplier; // i.e., AN1
    std::vector<Float_> gain_multiplier; // i.e., AN2
    std::vector<Float_> wcss_loss; // i.e., D
 
    std::vector<UpdateHistory<Index_> > update_history; // i.e., NCP, LIVE, and ITRAN. 
 
    Index_ optra_steps_since_last_transfer = 0; // i.e., INDX
 
public:
    Workspace(Index_ nobs, Cluster_ ncenters) :
        // Sizes taken from the .Fortran() call in stats::kmeans(). 
        best_destination_cluster(nobs), 
        cluster_sizes(ncenters),
        loss_multiplier(ncenters),
        gain_multiplier(ncenters),
        wcss_loss(nobs),
        update_history(ncenters)
    {}
};
 
template<typename Data_, typename Float_, typename Dim_>
Float_ squared_distance_from_cluster(const Data_* data, const Float_* center, Dim_ ndim) {
    Float_ output = 0;
    for (decltype(ndim) dim = 0; dim < ndim; ++dim, ++data, ++center) {
        Float_ delta = static_cast<Float_>(*data) - *center; // cast to float for consistent precision regardless of Data_.
        output += delta * delta;
    }
    return output;
}
 
template<class Matrix_, typename Cluster_, typename Float_>
void find_closest_two_centers(const Matrix_& data, Cluster_ ncenters, const Float_* centers, Cluster_* best_cluster, std::vector<Cluster_>& best_destination_cluster, int nthreads) {
    auto ndim = data.num_dimensions();
 
    // We assume that there are at least two centers here, otherwise we should
    // have detected that this was an edge case in RefineHartiganWong::run.
    internal::QuickSearch<Float_, Cluster_, decltype(ndim)> index(ndim, ncenters, centers);
 
    auto nobs = data.num_observations();
    typedef typename Matrix_::index_type Index_;
    parallelize(nthreads, nobs, [&](int, Index_ start, Index_ length) -> void {
        auto matwork = data.create_workspace(start, length);
        for (Index_ obs = start, end = start + length; obs < end; ++obs) {
            auto optr = data.get_observation(matwork);
            auto res2 = index.find2(optr);
            best_cluster[obs] = res2.first;
            best_destination_cluster[obs] = res2.second;
        }
    });
}
 
template<typename Float_>
constexpr Float_ big_number() {
    return 1e30; // Some very big number.
}
 
template<typename Dim_, typename Data_, typename Index_, typename Cluster_, typename Float_>
void transfer_point(Dim_ ndim, const Data_* obs_ptr, Index_ obs_id, Cluster_ l1, Cluster_ l2, Float_* centers, Cluster_* best_cluster, Workspace<Float_, Index_, Cluster_>& work) {
    // Yes, casts to float are deliberate here, so that the
    // multipliers can be computed correctly.
    Float_ al1 = work.cluster_sizes[l1], alw = al1 - 1;
    Float_ al2 = work.cluster_sizes[l2], alt = al2 + 1;
 
    size_t long_ndim = ndim;
    auto copy1 = centers + static_cast<size_t>(l1) * long_ndim; // cast to avoid overflow.
    auto copy2 = centers + static_cast<size_t>(l2) * long_ndim;
    for (decltype(ndim) dim = 0; dim < ndim; ++dim, ++copy1, ++copy2, ++obs_ptr) {
        Float_ oval = *obs_ptr; // cast to float for consistent precision regardless of Data_.
        *copy1 = (*copy1 * al1 - oval) / alw;
        *copy2 = (*copy2 * al2 + oval) / alt;
    }
 
    --work.cluster_sizes[l1];
    ++work.cluster_sizes[l2];
 
    work.gain_multiplier[l1] = alw / al1;
    work.loss_multiplier[l1] = (alw > 1 ? alw / (alw - 1) : big_number<Float_>());
    work.loss_multiplier[l2] = alt / al2;
    work.gain_multiplier[l2] = alt / (alt + 1);
 
    best_cluster[obs_id] = l2;
    work.best_destination_cluster[obs_id] = l1;
}
 
/* ALGORITHM AS 136.1  APPL. STATIST. (1979) VOL.28, NO.1
 * This is the OPtimal TRAnsfer stage.
 *             ----------------------
 * Each point is re-assigned, if necessary, to the cluster that will induce a
 * maximum reduction in the within-cluster sum of squares. In this stage,
 * there is only one pass through the data.
 */
template<class Matrix_, typename Cluster_, typename Float_>
bool optimal_transfer(const Matrix_& data, Workspace<Float_, typename Matrix_::index_type, Cluster_>& work, Cluster_ ncenters, Float_* centers, Cluster_* best_cluster, bool all_live) {
    auto nobs = data.num_observations();
    auto ndim = data.num_dimensions();
    auto matwork = data.create_workspace();
    size_t long_ndim = ndim;
 
    for (decltype(nobs) obs = 0; obs < nobs; ++obs) { 
        ++work.optra_steps_since_last_transfer;
 
        auto l1 = best_cluster[obs];
        if (work.cluster_sizes[l1] != 1) {
            auto obs_ptr = data.get_observation(obs, matwork);
 
            // The original Fortran implementation re-used the WCSS loss for
            // each observation, only recomputing it if its cluster had
            // experienced an optimal transfer for an earlier observation. In
            // theory, this sounds great to avoid recomputation, but the
            // existing WCSS loss was computed in a running fashion during the
            // quick transfers. This makes them susceptible to accumulation of
            // numerical errors, even after the centroids are freshly
            // recomputed in the run() loop. So, we simplify matters and
            // improve accuracy by just recomputing the loss all the time,
            // which doesn't take too much extra effort.
            auto& wcss_loss = work.wcss_loss[obs];
            auto l1_ptr = centers + long_ndim * static_cast<size_t>(l1); // cast to avoid overflow.
            wcss_loss = squared_distance_from_cluster(obs_ptr, l1_ptr, ndim) * work.loss_multiplier[l1];
 
            // Find the cluster with minimum WCSS gain.
            auto l2 = work.best_destination_cluster[obs];
            auto original_l2 = l2;
            auto l2_ptr = centers + long_ndim * static_cast<size_t>(l2); // cast to avoid overflow.
            auto wcss_gain = squared_distance_from_cluster(obs_ptr, l2_ptr, ndim) * work.gain_multiplier[l2];
 
            auto update_destination_cluster = [&](Cluster_ cen) -> void {
                auto cen_ptr = centers + long_ndim * static_cast<size_t>(cen); // cast to avoid overflow.
                auto candidate = squared_distance_from_cluster(obs_ptr, cen_ptr, ndim) * work.gain_multiplier[cen];
                if (candidate < wcss_gain) {
                    wcss_gain = candidate;
                    l2 = cen;
                }
            };
 
            // If the current assigned cluster is live, we need to check all
            // other clusters as potential transfer destinations, because the
            // gain/loss comparison has changed. Otherwise, we only need to
            // consider other live clusters as transfer destinations; the
            // non-live clusters were already rejected as transfer destinations
            // when compared to the current assigned cluster, so there's no
            // point checking them again if they didn't change in the meantime.
            //
            // The exception is for the first call to optimal_transfer, where we
            // consider all clusters as live (i.e., all_live = true). This is
            // because no observation really knows its best transfer
            // destination yet - the second-closest cluster is just a
            // guesstimate - so we need to compute it exhaustively.
            if (all_live || work.update_history[l1].is_live(obs)) { 
                for (Cluster_ cen = 0; cen < ncenters; ++cen) {
                    if (cen != l1 && cen != original_l2) {
                        update_destination_cluster(cen);
                    }
                }
            } else {
                for (Cluster_ cen = 0; cen < ncenters; ++cen) {
                    if (cen != l1 && cen != original_l2 && work.update_history[cen].is_live(obs)) {
                        update_destination_cluster(cen);
                    }
                }
            }
 
            // Deciding whether to make the transfer based on the change to the WCSS.
            if (wcss_gain >= wcss_loss) {
                work.best_destination_cluster[obs] = l2;
            } else {
                work.optra_steps_since_last_transfer = 0;
                work.update_history[l1].set_optimal(obs);
                work.update_history[l2].set_optimal(obs);
                transfer_point(ndim, obs_ptr, obs, l1, l2, centers, best_cluster, work);
            }
        }
 
        // Stop if we've iterated through the entire dataset and no transfer of
        // any kind took place, be it optimal or quick.
        if (work.optra_steps_since_last_transfer == nobs) {
            return true;
        }
    }
 
    return false;
} 
 
/* ALGORITHM AS 136.2  APPL. STATIST. (1979) VOL.28, NO.1 
 * This is the Quick TRANsfer stage. 
 *             -------------------- 
 * IC1(I) is the cluster which point I currently belongs to.
 * IC2(I) is the cluster which point I is most likely to be transferred to.
 *
 * For each point I, IC1(I) & IC2(I) are switched, if necessary, to reduce
 * within-cluster sum of squares. The cluster centres are updated after each
 * step. In this stage, we loop through the data until no further change is to
 * take place, or we hit an iteration limit, whichever is first.
 */
template<class Matrix_, typename Cluster_, typename Float_>
std::pair<bool, bool> quick_transfer(
    const Matrix_& data,
    Workspace<Float_, typename Matrix_::index_type, Cluster_>& work,
    Float_* centers,
    Cluster_* best_cluster,
    int quick_iterations)
{
    bool had_transfer = false;
 
    auto nobs = data.num_observations();
    auto matwork = data.create_workspace();
    auto ndim = data.num_dimensions();
    size_t long_ndim = data.num_dimensions();
 
    typedef decltype(nobs) Index_;
    Index_ steps_since_last_quick_transfer = 0; // i.e., ICOUN in the original Fortran implementation.
 
    for (int it = 0; it < quick_iterations; ++it) {
        int prev_it = it - 1;
 
        for (decltype(nobs) obs = 0; obs < nobs; ++obs) { 
            ++steps_since_last_quick_transfer;
            auto l1 = best_cluster[obs];
 
            if (work.cluster_sizes[l1] != 1) {
                const typename Matrix_::data_type* obs_ptr = NULL;
 
                // Need to update the WCSS loss if the cluster was updated recently. 
                // Otherwise, we must have already updated the WCSS in a previous 
                // iteration of the outermost loop, so this can be skipped.
                //
                // Note that we use changed_at_or_after; if the same
                // observation was changed in the previous iteration of the
                // outermost loop, its WCSS loss won't have been updated yet.
                auto& history1 = work.update_history[l1];
                if (history1.changed_after_or_at(prev_it, obs)) {
                    auto l1_ptr = centers + static_cast<size_t>(l1) * long_ndim; // cast to avoid overflow.
                    obs_ptr = data.get_observation(obs, matwork);
                    work.wcss_loss[obs] = squared_distance_from_cluster(obs_ptr, l1_ptr, ndim) * work.loss_multiplier[l1];
                }
 
                // If neither the best or second-best clusters have changed
                // after the previous iteration that we visited this
                // observation, then there's no point reevaluating the
                // transfer, because nothing's going to be different anyway.
                auto l2 = work.best_destination_cluster[obs];
                auto& history2 = work.update_history[l2];
                if (history1.changed_after(prev_it, obs) || history2.changed_after(prev_it, obs)) {
                    if (obs_ptr == NULL) {
                        obs_ptr = data.get_observation(obs, matwork);
                    }
                    auto l2_ptr = centers + static_cast<size_t>(l2) * long_ndim; // cast to avoid overflow.
                    auto wcss_gain = squared_distance_from_cluster(obs_ptr, l2_ptr, ndim) * work.gain_multiplier[l2];
 
                    if (wcss_gain < work.wcss_loss[obs]) {
                        had_transfer = true;
                        steps_since_last_quick_transfer = 0;
                        history1.set_quick(it, obs);
                        history2.set_quick(it, obs);
                        transfer_point(ndim, obs_ptr, obs, l1, l2, centers, best_cluster, work);
                    }
               }
           }
 
           if (steps_since_last_quick_transfer == nobs) {
               // Quit early if no transfer occurred within the past 'nobs'
               // steps, as we've already converged for each observation. 
               return std::make_pair(had_transfer, false);
           }
        }
    } 
 
    return std::make_pair(had_transfer, true);
}
 
}
template<typename Matrix_ = SimpleMatrix<double, int>, typename Cluster_ = int, typename Float_ = double>
class RefineHartiganWong : public Refine<Matrix_, Cluster_, Float_> {
public:
    RefineHartiganWong(RefineHartiganWongOptions options) : my_options(std::move(options)) {}
 
    RefineHartiganWong() = default;
 
private:
    RefineHartiganWongOptions my_options;
    typedef typename Matrix_::index_type Index_;
 
public:
    RefineHartiganWongOptions& get_options() {
        return my_options;
    }
 
public:
    Details<Index_> run(const Matrix_& data, Cluster_ ncenters, Float_* centers, Cluster_* clusters) const {
        auto nobs = data.num_observations();
        if (internal::is_edge_case(nobs, ncenters)) {
            return internal::process_edge_case(data, ncenters, centers, clusters);
        }
 
        RefineHartiganWong_internal::Workspace<Float_, Index_, Cluster_> work(nobs, ncenters);
 
        RefineHartiganWong_internal::find_closest_two_centers(data, ncenters, centers, clusters, work.best_destination_cluster, my_options.num_threads);
        for (Index_ obs = 0; obs < nobs; ++obs) {
            ++work.cluster_sizes[clusters[obs]];
        }
        internal::compute_centroids(data, ncenters, centers, clusters, work.cluster_sizes);
 
        for (Cluster_ cen = 0; cen < ncenters; ++cen) {
            Float_ num = work.cluster_sizes[cen]; // yes, cast is deliberate here so that the multipliers can be computed correctly.
            work.gain_multiplier[cen] = num / (num + 1);
            work.loss_multiplier[cen] = (num > 1 ? num / (num - 1) : RefineHartiganWong_internal::big_number<Float_>());
        }
 
        int iter = 0;
        int ifault = 0;
        while ((++iter) <= my_options.max_iterations) {
            bool finished = RefineHartiganWong_internal::optimal_transfer(data, work, ncenters, centers, clusters, /* all_live = */ (iter == 1));
            if (finished) {
                break;
            }
 
            auto quick_status = RefineHartiganWong_internal::quick_transfer(
                data,
                work,
                centers,
                clusters,
                my_options.max_quick_transfer_iterations
            );
 
            // Recomputing the centroids to avoid accumulation of numerical
            // errors after many transfers (e.g., adding a whole bunch of
            // values and then subtracting them again leaves behind some
            // cancellation error). Note that we don't have to do this if
            // 'finished = true' as this means that there was no transfer of
            // any kind in the final pass through the dataset.
            internal::compute_centroids(data, ncenters, centers, clusters, work.cluster_sizes);
 
            if (quick_status.second) { // Hit the quick transfer iteration limit.
                if (my_options.quit_on_quick_transfer_convergence_failure) {
                    ifault = 4;
                    break;
                }
            } else {
                // If quick transfer converged and there are only two clusters,
                // there is no need to re-enter the optimal transfer stage. 
                if (ncenters == 2) {
                    break;
                }
            }
 
            if (quick_status.first) { // At least one quick transfer was performed.
                work.optra_steps_since_last_transfer = 0;
            }
 
            for (Cluster_ c = 0; c < ncenters; ++c) {
                work.update_history[c].reset(nobs);
            }
        }
 
        if (iter == my_options.max_iterations + 1) {
            ifault = 2;
        }
 
        return Details(std::move(work.cluster_sizes), iter, ifault);
    }
};
 
}
 
#endif